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					<span class="ref toc here">Normal ordering for deformed boson operators and
operator-valued deformed Stirling numbers</span>
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						<li class="tocentry">
							<a href="#S1" title="§1. Introduction in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§1 Introduction</a>
						</li>
						<li class="tocentry">
							<a href="#S2" title="§2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§2 Stirling and deformed Stirling numbers</a>
						</li>
						<li class="tocentry">
							<a href="#S3" title="§3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§3 Some algebraic properties of deformed boson operators</a>
						</li>
						<li class="tocentry">
							<a href="#S4" title="§4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§4 Normal ordering of powers of the deformed number operator</a>
						</li>
						<li class="tocentry">
							<a href="#S5" title="§5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§5 Operator-valued deformed Stirling numbers</a>
						</li>
						<li class="tocentry">
							<a href="#S6" title="§6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§6 A generating function for the deformed Stirling numbers
of the first kind</a>
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						<li class="tocentry">
							<a href="#S7" title="§7. Discussion in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">§7 Discussion</a>
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						<li class="tocentry">
							<a href="#bib" title="References in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref toc">References</a>
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					<h1 class="title document-title">Normal ordering for deformed boson operators and
operator-valued deformed Stirling numbers</h1>
					<div class="creator author">
						<div class="personname"> Jacob Katriel
							<m:math display="inline">
								<m:semantics>
									<m:none/>
									<m:annotation-xml encoding="MathML-Content">
										<m:ci/>
									</m:annotation-xml>
								</m:semantics>
							</m:math> and Maurice Kibler <br class="break"/>
							<span style="" class="text slanted small">Institut de Physique Nucléaire de Lyon </span>
							<br class="break"/>
							<span style="" class="text slanted small">IN2P3-CNRS et Université Claude Bernard</span>
							<br class="break"/>
							<span style="" class="text slanted small">43 Boulevard du 11 Novembre 1918</span>
							<br class="break"/>
							<span style="" class="text slanted small">F-69622 Villeurbanne Cedex, France</span>
						</div>
  Permanent address:
Department of Chemistry, Technion - Israel Institute of Technology,
Haifa 32000, Israel. email: chr09kt@technion.email: Kibler@ipnl.in2p3.fr.</div>

					<div class="abstract">
						<h6>Abstract. </h6>

						<p class="p">The normal ordering formulae for powers of the boson number operator
							<m:math display="inline">
								<m:semantics>
									<m:mover accent="true">
										<m:mi>n</m:mi>
										<m:mo>⌃</m:mo>
									</m:mover>
									<m:annotation-xml encoding="MathML-Content">
										<m:apply>
											<m:ci>⌃</m:ci>
											<m:ci>n</m:ci>
										</m:apply>
									</m:annotation-xml>
								</m:semantics>
							</m:math>
are extended to deformed bosons. It is found that for the “M-type”
deformed bosons, which satisfy <m:math display="inline">
								<m:semantics>
									<m:mrow>
										<m:mrow>
											<m:mrow>
												<m:mi>a</m:mi>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mo>†</m:mo>
												</m:msup>
											</m:mrow>
											<m:mo>-</m:mo>
											<m:mrow>
												<m:mi>q</m:mi>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mo>†</m:mo>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:mi>a</m:mi>
											</m:mrow>
										</m:mrow>
										<m:mo>=</m:mo>
										<m:mn>1</m:mn>
									</m:mrow>
									<m:annotation-xml encoding="MathML-Content">
										<m:apply>
											<m:eq/>
											<m:apply>
												<m:minus/>
												<m:apply>
													<m:times/>
													<m:ci>a</m:ci>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:times/>
													<m:ci>q</m:ci>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
													<m:ci>a</m:ci>
												</m:apply>
											</m:apply>
											<m:cn>1</m:cn>
										</m:apply>
									</m:annotation-xml>
								</m:semantics>
							</m:math>,
the extension involves a set of deformed Stirling numbers which
replace the Stirling numbers occurring in the conventional case.
On the other hand, the deformed Stirling numbers which
have to be introduced in the case of the “P-type” deformed bosons,
which satisfy <m:math display="inline">
								<m:semantics>
									<m:mrow>
										<m:mrow>
											<m:mrow>
												<m:mi>a</m:mi>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mo>†</m:mo>
												</m:msup>
											</m:mrow>
											<m:mo>-</m:mo>
											<m:mrow>
												<m:mi>q</m:mi>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mo>†</m:mo>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:mi>a</m:mi>
											</m:mrow>
										</m:mrow>
										<m:mo>=</m:mo>
										<m:msup>
											<m:mi>q</m:mi>
											<m:mrow>
												<m:mo>-</m:mo>
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:mrow>
										</m:msup>
									</m:mrow>
									<m:annotation-xml encoding="MathML-Content">
										<m:apply>
											<m:eq/>
											<m:apply>
												<m:minus/>
												<m:apply>
													<m:times/>
													<m:ci>a</m:ci>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:times/>
													<m:ci>q</m:ci>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
													<m:ci>a</m:ci>
												</m:apply>
											</m:apply>
											<m:apply>
												<m:csymbol cd="ambiguous">superscript</m:csymbol>
												<m:ci>q</m:ci>
												<m:apply>
													<m:minus/>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:apply>
									</m:annotation-xml>
								</m:semantics>
							</m:math>,
are found to depend on the operator <m:math display="inline">
								<m:semantics>
									<m:mover accent="true">
										<m:mi>n</m:mi>
										<m:mo>⌃</m:mo>
									</m:mover>
									<m:annotation-xml encoding="MathML-Content">
										<m:apply>
											<m:ci>⌃</m:ci>
											<m:ci>n</m:ci>
										</m:apply>
									</m:annotation-xml>
								</m:semantics>
							</m:math>. This distinction
between the two types of deformed bosons is in harmony with earlier
observations made in the context of a study of the extended
Campbell-Baker-Hausdorff formula.</p>

						<p class="p">Published in J. Phys. A : Math. Gen. 25 (1992) 2683-2691.</p>

					</div>

					<div class="section" id="S1">
						<h2 class="title section-title">
							<span class="refnum">
								<span class="reftype">§ </span>1. </span>Introduction</h2>
						<div class="para" id="S1.p1">
							<p class="p">The transformation of a second-quantized operator
into a normally ordered form, in which each
term is written with the creation operators preceding the annihilation
operators, has been found to simplify quantum mechanical calculations in
a large and varied range of situations. Techniques for the
accomplishment of this ordering  have been developed and are widely
utilized [1,2]. A particular
subclass of problems and techniques involves situations in which the
operators of interest commute with the number operator.
More specifically,
one is interested in transforming an operator  which is  a function  of
the number operator into a normally ordered form,
or transforming an operator
each of whose terms has an equal number of creation and annihilation
operators corresponding to each degree of freedom, into an equivalent
operator expressed in terms of the number operator only.</p>
						</div>

						<div class="para" id="S1.p2">
							<p class="p">In the present article we consider the corresponding problem for the
deformed bosons which have been investigated very extensively in the
last three years [3,4]
in connection with the recent interest in
the properties and
applications of quantum groups.</p>
						</div>

					</div>

					<div class="section" id="S2">
						<h2 class="title section-title">
							<span class="refnum">
								<span class="reftype">§ </span>2. </span>Stirling and deformed Stirling numbers</h2>
						<div class="para" id="S2.p1">
							<p class="p">The Stirling numbers of the first <m:math display="inline">
									<m:semantics>
										<m:mfenced open="(" close=")">
											<m:mi>s</m:mi>
										</m:mfenced>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>s</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
and second <m:math display="inline">
									<m:semantics>
										<m:mfenced open="(" close=")">
											<m:mi>S</m:mi>
										</m:mfenced>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>S</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> kinds were introduced in
connection with the expression for a descending product of a
variable <m:math display="inline">
									<m:semantics>
										<m:mi>x</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>x</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> as a linear combination of integral and positive powers of
that variable, and the inverse relation, respectively <cite class="cite">[<a href="#bib.bib5" title="" class="ref">5</a>]</cite>
							</p>
							<table class="equation" id="S2.E1">


								<tr valign="baseline" class="equation" id="S2.E1">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mi>x</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>x</m:mi>
																<m:mo>-</m:mo>
																<m:mn>1</m:mn>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:mi mathvariant="normal">⋯</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>x</m:mi>
																<m:mo>-</m:mo>
																<m:mi>k</m:mi>
																<m:mo>+</m:mo>
																<m:mn>1</m:mn>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mover>
															<m:munder>
																<m:mo movablelimits="false">∑</m:mo>
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo movablelimits="false">=</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:munder>
															<m:mi>k</m:mi>
														</m:mover>
														<m:mi>s</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>x</m:mi>
															<m:mi>m</m:mi>
														</m:msup>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>x</m:ci>
															<m:apply>
																<m:minus/>
																<m:ci>x</m:ci>
																<m:cn>1</m:cn>
															</m:apply>
															<m:ci>⋯</m:ci>
															<m:apply>
																<m:plus/>
																<m:apply>
																	<m:minus/>
																	<m:ci>x</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:cn>1</m:cn>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>s</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>x</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">1</span>)</td>
								</tr>

							</table>
							<table class="equation" id="S2.E2">


								<tr valign="baseline" class="equation" id="S2.E2">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msup>
															<m:mi>x</m:mi>
															<m:mi>m</m:mi>
														</m:msup>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>m</m:mi>
															</m:mover>
															<m:mi>S</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:mi>x</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>x</m:mi>
																	<m:mo>-</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:mi mathvariant="normal">⋯</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>x</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>+</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>x</m:ci>
															<m:ci>m</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>S</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:ci>x</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:ci>x</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>⋯</m:ci>
																<m:apply>
																	<m:plus/>
																	<m:apply>
																		<m:minus/>
																		<m:ci>x</m:ci>
																		<m:ci>k</m:ci>
																	</m:apply>
																	<m:cn>1</m:cn>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">2</span>)</td>
								</tr>

							</table>
							<p class="p">Using these defining relations it is easy to show that the Stirling
numbers satisfy the recurrence relations</p>
							<table class="equation" id="S2.E3">


								<tr valign="baseline" class="equation" id="S2.E3">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mi>s</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>+</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mrow>
															<m:mi>s</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>-</m:mo>
														<m:mrow>
															<m:mi>k</m:mi>
															<m:mo>⁢</m:mo>
															<m:mi>s</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>m</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>s</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>k</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:times/>
																<m:ci>s</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>k</m:ci>
																<m:ci>s</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">3</span>)</td>
								</tr>

							</table>
							<p class="p">and</p>
							<table class="equation" id="S2.E4">


								<tr valign="baseline" class="equation" id="S2.E4">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mi>S</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mrow>
																<m:mi>S</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo>+</m:mo>
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>⁢</m:mo>
																<m:mi>S</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>k</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo>,</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>S</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>m</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:ci>S</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>k</m:ci>
																<m:ci>S</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">4</span>)</td>
								</tr>

							</table>
							<p class="p">with the initial values <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi>s</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mn>1</m:mn>
														<m:mo>,</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mi>S</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mn>1</m:mn>
														<m:mo>,</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci/>
												<m:apply>
													<m:times/>
													<m:ci>s</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:cn>1</m:cn>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:ci>S</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:cn>1</m:cn>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
												<m:eq/>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
and the “boundary conditions”
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi>s</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>i</m:mi>
														<m:mo>,</m:mo>
														<m:mi>j</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mi>S</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>i</m:mi>
														<m:mo>,</m:mo>
														<m:mi>j</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mn>0</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci/>
												<m:apply>
													<m:times/>
													<m:ci>s</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>i</m:ci>
														<m:ci>j</m:ci>
													</m:apply>
												</m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:ci>S</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>i</m:ci>
														<m:ci>j</m:ci>
													</m:apply>
												</m:apply>
												<m:eq/>
												<m:cn>0</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> for
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>i</m:mi>
											<m:mo>&lt;</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:lt/>
												<m:ci>i</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>j</m:mi>
											<m:mo>&lt;</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:lt/>
												<m:ci>j</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and for <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>i</m:mi>
											<m:mo>&lt;</m:mo>
											<m:mi>j</m:mi>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:lt/>
												<m:ci>i</m:ci>
												<m:ci>j</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
The combinatorial significance of the Stirling numbers has been amply
discussed <cite class="cite">[<a href="#bib.bib6" title="" class="ref">6</a>]</cite>.</p>
						</div>

						<div class="para" id="S2.p2">
							<p class="p">Several generalisations of the Stirling numbers appeared in the
mathematical literature [7-12].
In anticipation of further development we shall refer to them
generically as deformed Stirling numbers. In this context
we wish to distinguish between
the two widely used forms of “deformed numbers”
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>x</m:mi>
												</m:mfenced>
												<m:mi>M</m:mi>
											</m:msub>
											<m:mo>=</m:mo>
											<m:mfrac>
												<m:mrow>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mi>x</m:mi>
													</m:msup>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
												<m:mrow>
													<m:mi>q</m:mi>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:mfrac>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>x</m:ci>
													<m:ci>M</m:ci>
												</m:apply>
												<m:apply>
													<m:divide/>
													<m:apply>
														<m:minus/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:ci>x</m:ci>
														</m:apply>
														<m:cn>1</m:cn>
													</m:apply>
													<m:apply>
														<m:minus/>
														<m:ci>q</m:ci>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, the usual choice in the
mathematical literature on <m:math display="inline">
									<m:semantics>
										<m:mi>q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-analysis <cite class="cite">[<a href="#bib.bib13" title="" class="ref">13</a>]</cite>, and
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>x</m:mi>
												</m:mfenced>
												<m:mi>P</m:mi>
											</m:msub>
											<m:mo>=</m:mo>
											<m:mfrac>
												<m:mrow>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mi>x</m:mi>
													</m:msup>
													<m:mo>-</m:mo>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mrow>
															<m:mo>-</m:mo>
															<m:mi>x</m:mi>
														</m:mrow>
													</m:msup>
												</m:mrow>
												<m:mrow>
													<m:mi>q</m:mi>
													<m:mo>-</m:mo>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mrow>
															<m:mo>-</m:mo>
															<m:mn>1</m:mn>
														</m:mrow>
													</m:msup>
												</m:mrow>
											</m:mfrac>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>x</m:ci>
													<m:ci>P</m:ci>
												</m:apply>
												<m:apply>
													<m:divide/>
													<m:apply>
														<m:minus/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:ci>x</m:ci>
														</m:apply>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:apply>
																<m:minus/>
																<m:ci>x</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:minus/>
														<m:ci>q</m:ci>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:apply>
																<m:minus/>
																<m:cn>1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
which is common to the recent physical literature and to
the literature on quantum groups.
A generalisation was recently proposed by
Wachs and White <cite class="cite">[<a href="#bib.bib12" title="" class="ref">12</a>]</cite>, which can be written in the form
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>x</m:mi>
												</m:mfenced>
												<m:mi>G</m:mi>
											</m:msub>
											<m:mo>=</m:mo>
											<m:mfrac>
												<m:mrow>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mi>x</m:mi>
													</m:msup>
													<m:mo>-</m:mo>
													<m:msup>
														<m:mi>p</m:mi>
														<m:mi>x</m:mi>
													</m:msup>
												</m:mrow>
												<m:mrow>
													<m:mi>q</m:mi>
													<m:mo>-</m:mo>
													<m:mi>p</m:mi>
												</m:mrow>
											</m:mfrac>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>x</m:ci>
													<m:ci>G</m:ci>
												</m:apply>
												<m:apply>
													<m:divide/>
													<m:apply>
														<m:minus/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:ci>x</m:ci>
														</m:apply>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>p</m:ci>
															<m:ci>x</m:ci>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:minus/>
														<m:ci>q</m:ci>
														<m:ci>p</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
This form contains <m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mi>x</m:mi>
											</m:mfenced>
											<m:mi>M</m:mi>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:ci>x</m:ci>
												<m:ci>M</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mi>x</m:mi>
											</m:mfenced>
											<m:mi>P</m:mi>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:ci>x</m:ci>
												<m:ci>P</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> as special cases,
corresponding to the choices <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mrow>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:minus/>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, respectively.
We shall write <m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mi>x</m:mi>
											</m:mfenced>
											<m:mrow>
												<m:mi>M</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mi>q</m:mi>
												</m:mfenced>
											</m:mrow>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:ci>x</m:ci>
												<m:apply>
													<m:times/>
													<m:ci>M</m:ci>
													<m:ci>q</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, <m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mi>x</m:mi>
											</m:mfenced>
											<m:mrow>
												<m:mi>P</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mi>q</m:mi>
												</m:mfenced>
											</m:mrow>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:ci>x</m:ci>
												<m:apply>
													<m:times/>
													<m:ci>P</m:ci>
													<m:ci>q</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and
								<m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mi>x</m:mi>
											</m:mfenced>
											<m:mrow>
												<m:mi>G</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>p</m:mi>
														<m:mo>,</m:mo>
														<m:mi>q</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:ci>x</m:ci>
												<m:apply>
													<m:times/>
													<m:ci>G</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>p</m:ci>
														<m:ci>q</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> instead of the symbols introduced above
whenever the choice of the parameters <m:math display="inline">
									<m:semantics>
										<m:mi>q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and/or <m:math display="inline">
									<m:semantics>
										<m:mi>p</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>p</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> will
have to be explicated.
The identities</p>
							<table class="equation" id="S2.E5">


								<tr valign="baseline" class="equation" id="S2.E5">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msub>
														<m:mfenced open="[" close="]">
															<m:mi>x</m:mi>
														</m:mfenced>
														<m:mrow>
															<m:mi>P</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mi>q</m:mi>
															</m:mfenced>
														</m:mrow>
													</m:msub>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msup>
															<m:mi>q</m:mi>
															<m:mrow>
																<m:mo>-</m:mo>
																<m:mi>x</m:mi>
																<m:mo>+</m:mo>
																<m:mn>1</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mrow>
																<m:mi>M</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>q</m:mi>
																		<m:mn>2</m:mn>
																	</m:msup>
																</m:mfenced>
															</m:mrow>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mrow>
																<m:mi>G</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>p</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>q</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:msub>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msup>
															<m:mi>p</m:mi>
															<m:mrow>
																<m:mi>x</m:mi>
																<m:mo>-</m:mo>
																<m:mn>1</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mrow>
																<m:mi>M</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>q</m:mi>
																		<m:mo>/</m:mo>
																		<m:mi>p</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mrow>
																<m:mi>G</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>p</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>q</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:msub>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msup>
															<m:mfenced open="(" close=")">
																<m:msqrt>
																	<m:mrow>
																		<m:mi>p</m:mi>
																		<m:mo>⁢</m:mo>
																		<m:mi>q</m:mi>
																	</m:mrow>
																</m:msqrt>
															</m:mfenced>
															<m:mrow>
																<m:mi>x</m:mi>
																<m:mo>-</m:mo>
																<m:mn>1</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mrow>
																<m:mi>P</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:msqrt>
																		<m:mrow>
																			<m:mi>q</m:mi>
																			<m:mo>/</m:mo>
																			<m:mi>p</m:mi>
																		</m:mrow>
																	</m:msqrt>
																</m:mfenced>
															</m:mrow>
														</m:msub>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:ci>x</m:ci>
															<m:apply>
																<m:times/>
																<m:ci>P</m:ci>
																<m:ci>q</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:plus/>
																	<m:apply>
																		<m:minus/>
																		<m:ci>x</m:ci>
																	</m:apply>
																	<m:cn>1</m:cn>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:apply>
																	<m:times/>
																	<m:ci>M</m:ci>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>q</m:ci>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:apply>
																	<m:times/>
																	<m:ci>G</m:ci>
																	<m:apply>
																		<m:interval closure="open"/>
																		<m:ci>p</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>p</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:ci>x</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:apply>
																	<m:times/>
																	<m:ci>M</m:ci>
																	<m:apply>
																		<m:divide/>
																		<m:ci>q</m:ci>
																		<m:ci>p</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:apply>
																	<m:times/>
																	<m:ci>G</m:ci>
																	<m:apply>
																		<m:interval closure="open"/>
																		<m:ci>p</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:ci/>
																	<m:apply>
																		<m:times/>
																		<m:ci>p</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:minus/>
																	<m:ci>x</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:apply>
																	<m:times/>
																	<m:ci>P</m:ci>
																	<m:apply>
																		<m:ci/>
																		<m:apply>
																			<m:divide/>
																			<m:ci>q</m:ci>
																			<m:ci>p</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">5</span>)</td>
								</tr>

							</table>
							<p class="p">illustrate the notation and exhibit
some of the elementary properties of these deformed numbers.</p>
						</div>

						<div class="para" id="S2.p3">
							<p class="p">One of the generalisations of the Stirling numbers <cite class="cite">[<a href="#bib.bib10" title="" class="ref">10</a>]</cite> involves
a descending product of M-type deformed
numbers expressed in terms of the powers
of the M-type deformed number <m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mi>x</m:mi>
											</m:mfenced>
											<m:mi>M</m:mi>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:ci>x</m:ci>
												<m:ci>M</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
							</p>
							<table class="equation" id="S2.E6">


								<tr valign="baseline" class="equation" id="S2.E6">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mi>M</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mi>x</m:mi>
																	<m:mo>-</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mfenced>
															<m:mi>M</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mi mathvariant="normal">⋯</m:mi>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mi>x</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>+</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mfenced>
															<m:mi>M</m:mi>
														</m:msub>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mover>
															<m:munder>
																<m:mo movablelimits="false">∑</m:mo>
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo movablelimits="false">=</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:munder>
															<m:mi>k</m:mi>
														</m:mover>
														<m:msub>
															<m:mi>s</m:mi>
															<m:mi>q</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mi>M</m:mi>
															<m:mi>m</m:mi>
														</m:msubsup>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:minus/>
																	<m:ci>x</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
															<m:ci>⋯</m:ci>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:plus/>
																	<m:apply>
																		<m:minus/>
																		<m:ci>x</m:ci>
																		<m:ci>k</m:ci>
																	</m:apply>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>s</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>x</m:ci>
																		<m:ci>M</m:ci>
																	</m:apply>
																	<m:ci>m</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">6</span>)</td>
								</tr>

							</table>
							<p class="p">and the corresponding inverse relation</p>
							<table class="equation" id="S2.E7">


								<tr valign="baseline" class="equation" id="S2.E7">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mi>M</m:mi>
															<m:mi>m</m:mi>
														</m:msubsup>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>m</m:mi>
															</m:mover>
															<m:msub>
																<m:mi>S</m:mi>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>x</m:mi>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mrow>
																		<m:mi>x</m:mi>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mi mathvariant="normal">⋯</m:mi>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mrow>
																		<m:mi>x</m:mi>
																		<m:mo>-</m:mo>
																		<m:mi>k</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>x</m:ci>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:minus/>
																		<m:ci>x</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:ci>⋯</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:plus/>
																		<m:apply>
																			<m:minus/>
																			<m:ci>x</m:ci>
																			<m:ci>k</m:ci>
																		</m:apply>
																		<m:cn>1</m:cn>
																	</m:apply>
																	<m:ci>M</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">7</span>)</td>
								</tr>

							</table>
							<p class="p">Using the defining relations it is easy
to show that the deformed Stirling numbers <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mi>s</m:mi>
												<m:mi>q</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mi>m</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>s</m:ci>
													<m:ci>q</m:ci>
												</m:apply>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>k</m:ci>
													<m:ci>m</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mi>S</m:mi>
												<m:mi>q</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>S</m:ci>
													<m:ci>q</m:ci>
												</m:apply>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
which are referred to in the mathematical literature as <m:math display="inline">
									<m:semantics>
										<m:mi>q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-Stirling
numbers of the first and second kind, respectively,
satisfy the recurrence relations</p>
							<table class="equation" id="S2.E8">


								<tr valign="baseline" class="equation" id="S2.E8">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mi>s</m:mi>
															<m:mi>q</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>+</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msup>
															<m:mi>q</m:mi>
															<m:mrow>
																<m:mo>-</m:mo>
																<m:mi>k</m:mi>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:none/>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:msub>
																		<m:mi>s</m:mi>
																		<m:mi>q</m:mi>
																	</m:msub>
																	<m:mo>⁢</m:mo>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>,</m:mo>
																			<m:mrow>
																				<m:mi>m</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
																<m:mo>-</m:mo>
																<m:mrow>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mi>k</m:mi>
																		</m:mfenced>
																		<m:mi>M</m:mi>
																	</m:msub>
																	<m:mo>⁢</m:mo>
																	<m:msub>
																		<m:mi>s</m:mi>
																		<m:mi>q</m:mi>
																	</m:msub>
																	<m:mo>⁢</m:mo>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>,</m:mo>
																			<m:mi>m</m:mi>
																		</m:mrow>
																	</m:mfenced>
																	<m:mo>⁢</m:mo>
																	<m:none/>
																</m:mrow>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>s</m:ci>
																<m:ci>q</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>k</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
															<m:ci/>
															<m:apply>
																<m:minus/>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>s</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:interval closure="open"/>
																		<m:ci>k</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>m</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>k</m:ci>
																		<m:ci>M</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>s</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:interval closure="open"/>
																		<m:ci>k</m:ci>
																		<m:ci>m</m:ci>
																	</m:apply>
																	<m:ci/>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">8</span>)</td>
								</tr>

							</table>
							<p class="p">and</p>
							<table class="equation" id="S2.E9">


								<tr valign="baseline" class="equation" id="S2.E9">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msub>
																<m:mi>S</m:mi>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mrow>
																<m:msup>
																	<m:mi>q</m:mi>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mi>S</m:mi>
																	<m:mi>q</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo>+</m:mo>
															<m:mrow>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>k</m:mi>
																	</m:mfenced>
																	<m:mi>M</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mi>S</m:mi>
																	<m:mi>q</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>k</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo>,</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>S</m:ci>
																<m:ci>q</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>m</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>k</m:ci>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">9</span>)</td>
								</tr>

							</table>
							<p class="p">with “boundary conditions” and initial values identical with those
specified above for the conventional Stirling numbers.</p>
						</div>

						<div class="para" id="S2.p4">
							<p class="p">A slight modification in the form of the descending product, replacing
the factors <m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mrow>
													<m:mi>x</m:mi>
													<m:mo>-</m:mo>
													<m:mi>i</m:mi>
												</m:mrow>
											</m:mfenced>
											<m:mi>M</m:mi>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:apply>
													<m:minus/>
													<m:ci>x</m:ci>
													<m:ci>i</m:ci>
												</m:apply>
												<m:ci>M</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> by <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>x</m:mi>
												</m:mfenced>
												<m:mi>M</m:mi>
											</m:msub>
											<m:mo>-</m:mo>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>i</m:mi>
												</m:mfenced>
												<m:mi>M</m:mi>
											</m:msub>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:minus/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>x</m:ci>
													<m:ci>M</m:ci>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>i</m:ci>
													<m:ci>M</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, results in the relations [8-10]</p>
							<table class="equation" id="S2.E10">


								<tr valign="baseline" class="equation" id="S2.E10">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mi>M</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>x</m:mi>
																	</m:mfenced>
																	<m:mi>M</m:mi>
																</m:msub>
																<m:mo>-</m:mo>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mn>1</m:mn>
																	</m:mfenced>
																	<m:mi>M</m:mi>
																</m:msub>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:mi mathvariant="normal">⋯</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>x</m:mi>
																	</m:mfenced>
																	<m:mi>M</m:mi>
																</m:msub>
																<m:mo>-</m:mo>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mfenced>
																	<m:mi>M</m:mi>
																</m:msub>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mover>
															<m:munder>
																<m:mo movablelimits="false">∑</m:mo>
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo movablelimits="false">=</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:munder>
															<m:mi>k</m:mi>
														</m:mover>
														<m:msub>
															<m:mover accent="true">
																<m:mi>s</m:mi>
																<m:mo>~</m:mo>
															</m:mover>
															<m:mi>q</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mi>M</m:mi>
															<m:mi>m</m:mi>
														</m:msubsup>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
															<m:apply>
																<m:minus/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>x</m:ci>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:cn>1</m:cn>
																	<m:ci>M</m:ci>
																</m:apply>
															</m:apply>
															<m:ci>⋯</m:ci>
															<m:apply>
																<m:minus/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>x</m:ci>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																	<m:ci>M</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>~</m:ci>
																		<m:ci>s</m:ci>
																	</m:apply>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>x</m:ci>
																		<m:ci>M</m:ci>
																	</m:apply>
																	<m:ci>m</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">10</span>)</td>
								</tr>

							</table>
							<p class="p">and</p>
							<table class="equation" id="S2.E11">


								<tr valign="baseline" class="equation" id="S2.E11">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mi>x</m:mi>
															</m:mfenced>
															<m:mi>M</m:mi>
															<m:mi>m</m:mi>
														</m:msubsup>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>m</m:mi>
															</m:mover>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>~</m:mo>
																</m:mover>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>x</m:mi>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mi>x</m:mi>
																		</m:mfenced>
																		<m:mi>M</m:mi>
																	</m:msub>
																	<m:mo>-</m:mo>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mn>1</m:mn>
																		</m:mfenced>
																		<m:mi>M</m:mi>
																	</m:msub>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:mi mathvariant="normal">⋯</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mi>x</m:mi>
																		</m:mfenced>
																		<m:mi>M</m:mi>
																	</m:msub>
																	<m:mo>-</m:mo>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mrow>
																				<m:mi>k</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																		<m:mi>M</m:mi>
																	</m:msub>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>x</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>~</m:ci>
																		<m:ci>S</m:ci>
																	</m:apply>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>x</m:ci>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:apply>
																	<m:minus/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>x</m:ci>
																		<m:ci>M</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:cn>1</m:cn>
																		<m:ci>M</m:ci>
																	</m:apply>
																</m:apply>
																<m:ci>⋯</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>x</m:ci>
																		<m:ci>M</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:apply>
																			<m:minus/>
																			<m:ci>k</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																		<m:ci>M</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">11</span>)</td>
								</tr>

							</table>
							<p class="p">Starting with these defining relations and using the identity
								<cite class="cite">[<a href="#bib.bib9" title="" class="ref">9</a>]</cite>
							</p>
							<table class="equation" id="S2.E12">


								<tr valign="baseline" class="equation" id="S2.E12">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>a</m:mi>
															</m:mfenced>
															<m:mi>M</m:mi>
														</m:msub>
														<m:mo>-</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>b</m:mi>
															</m:mfenced>
															<m:mi>M</m:mi>
														</m:msub>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msup>
															<m:mi>q</m:mi>
															<m:mi>b</m:mi>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mi>a</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>b</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mi>M</m:mi>
														</m:msub>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>b</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:ci>b</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:minus/>
																	<m:ci>a</m:ci>
																	<m:ci>b</m:ci>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">12</span>)</td>
								</tr>

							</table>
							<p class="p">we obtain the recurrence relations</p>
							<table class="equation" id="S2.E13">


								<tr valign="baseline" class="equation" id="S2.E13">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mover accent="true">
																<m:mi>s</m:mi>
																<m:mo>~</m:mo>
															</m:mover>
															<m:mi>q</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>+</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mrow>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>s</m:mi>
																	<m:mo>~</m:mo>
																</m:mover>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>-</m:mo>
														<m:mrow>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>k</m:mi>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>s</m:mi>
																	<m:mo>~</m:mo>
																</m:mover>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>m</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>~</m:ci>
																	<m:ci>s</m:ci>
																</m:apply>
																<m:ci>q</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>k</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>~</m:ci>
																		<m:ci>s</m:ci>
																	</m:apply>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>k</m:ci>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>~</m:ci>
																		<m:ci>s</m:ci>
																	</m:apply>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">13</span>)</td>
								</tr>

							</table>
							<p class="p">and</p>
							<table class="equation" id="S2.E14">


								<tr valign="baseline" class="equation" id="S2.E14">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>~</m:mo>
																</m:mover>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mrow>
																<m:msub>
																	<m:mover accent="true">
																		<m:mi>S</m:mi>
																		<m:mo>~</m:mo>
																	</m:mover>
																	<m:mi>q</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo>+</m:mo>
															<m:mrow>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>k</m:mi>
																	</m:mfenced>
																	<m:mi>M</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mover accent="true">
																		<m:mi>S</m:mi>
																		<m:mo>~</m:mo>
																	</m:mover>
																	<m:mi>q</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>k</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo>,</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>~</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:ci>q</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>m</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>~</m:ci>
																		<m:ci>S</m:ci>
																	</m:apply>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>k</m:ci>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>~</m:ci>
																		<m:ci>S</m:ci>
																	</m:apply>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">14</span>)</td>
								</tr>

							</table>
							<p class="p">where the “boundary conditions” and initial values are, again,
as above. Note that</p>
							<table class="equation" id="S2.E15">


								<tr valign="baseline" class="equation" id="S2.E15">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>s</m:mi>
																	<m:mo>~</m:mo>
																</m:mover>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>m</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mrow>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>k</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>/</m:mo>
																	<m:mn>2</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mi>s</m:mi>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>m</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>~</m:mo>
																</m:mover>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mrow>
																	<m:mo>-</m:mo>
																	<m:mrow>
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>⁢</m:mo>
																			<m:mfenced open="(" close=")">
																				<m:mrow>
																					<m:mi>k</m:mi>
																					<m:mo>-</m:mo>
																					<m:mn>1</m:mn>
																				</m:mrow>
																			</m:mfenced>
																		</m:mrow>
																		<m:mo>/</m:mo>
																		<m:mn>2</m:mn>
																	</m:mrow>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mi>S</m:mi>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>~</m:ci>
																	<m:ci>s</m:ci>
																</m:apply>
																<m:ci>q</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:ci>m</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:divide/>
																	<m:apply>
																		<m:times/>
																		<m:ci>k</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>k</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:cn>2</m:cn>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>s</m:ci>
																<m:ci>q</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>~</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:ci>q</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>m</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:ci>k</m:ci>
																			<m:apply>
																				<m:minus/>
																				<m:ci>k</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>S</m:ci>
																<m:ci>q</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>m</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">15</span>)</td>
								</tr>

							</table>
							<p class="p">The two sets of deformed Stirling numbers of the first and second
kinds,
as well as the conventional Stirling numbers to which they
reduce in the limit <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>q</m:mi>
											<m:mo>→</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>→</m:ci>
												<m:ci>q</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, satisfy the following
dual relations</p>
							<table class="equation" id="S2.E16">


								<tr valign="baseline" class="equation" id="S2.E16">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mover>
															<m:munder>
																<m:mo movablelimits="false">∑</m:mo>
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo movablelimits="false">=</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:munder>
															<m:mi>k</m:mi>
														</m:mover>
														<m:msub>
															<m:mi>s</m:mi>
															<m:mi>q</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mi>S</m:mi>
															<m:mi>q</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>m</m:mi>
																<m:mo>,</m:mo>
																<m:msup>
																	<m:mi>k</m:mi>
																	<m:mo>′</m:mo>
																</m:msup>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mi>δ</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:msup>
																	<m:mi>k</m:mi>
																	<m:mo>′</m:mo>
																</m:msup>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>s</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>k</m:ci>
																		<m:ci>′</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:ci>δ</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>k</m:ci>
																	<m:ci>′</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">16</span>)</td>
								</tr>

							</table>
							<p class="p">and</p>
							<table class="equation" id="S2.E17">


								<tr valign="baseline" class="equation" id="S2.E17">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>m</m:mi>
															</m:mover>
															<m:msub>
																<m:mi>S</m:mi>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mi>s</m:mi>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:msup>
																		<m:mi>m</m:mi>
																		<m:mo>′</m:mo>
																	</m:msup>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mi>δ</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:msup>
																		<m:mi>m</m:mi>
																		<m:mo>′</m:mo>
																	</m:msup>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>s</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>m</m:ci>
																		<m:ci>′</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:ci>δ</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>m</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>m</m:ci>
																	<m:ci>′</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">17</span>)</td>
								</tr>

							</table>
							<p class="p">An additional set of deformed Stirling numbers of the second kind
was recently introduced by Wachs and White <cite class="cite">[<a href="#bib.bib12" title="" class="ref">12</a>]</cite>.
Their definition is motivated by combinatorial
considerations and has no algebraic origin. Their recurrence
relation reads</p>
							<table class="equation" id="S2.E18">


								<tr valign="baseline" class="equation" id="S2.E18">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mi>S</m:mi>
															<m:mrow>
																<m:mi>p</m:mi>
																<m:mo>,</m:mo>
																<m:mi>q</m:mi>
															</m:mrow>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>+</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
																<m:mo>,</m:mo>
																<m:mi>k</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mrow>
															<m:msup>
																<m:mi>p</m:mi>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>-</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mi>S</m:mi>
																<m:mrow>
																	<m:mi>p</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>q</m:mi>
																</m:mrow>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>+</m:mo>
														<m:mrow>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>k</m:mi>
																</m:mfenced>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mi>S</m:mi>
																<m:mrow>
																	<m:mi>p</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>q</m:mi>
																</m:mrow>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>S</m:ci>
																<m:apply>
																	<m:list/>
																	<m:ci>p</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>m</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:apply>
																		<m:list/>
																		<m:ci>p</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>k</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:apply>
																		<m:list/>
																		<m:ci>p</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">18</span>)</td>
								</tr>

							</table>
							<p class="p">and it reduces to (<a href="#S2.E14" title="Eq.14 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">14</a>) for <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

					</div>

					<div class="section" id="S3">
						<h2 class="title section-title">
							<span class="refnum">
								<span class="reftype">§ </span>3. </span>Some algebraic properties of deformed boson operators</h2>
						<div class="para" id="S3.p1">
							<p class="p">In the context of recent interest in quantum groups and their
realization, three types of
deformed boson operators have
been introduced [3,4,14]. The most straightforward definition starts by
postulating a Fock space on which creation (<m:math display="inline">
									<m:semantics>
										<m:mi>a</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>a</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>), annihilation
(<m:math display="inline">
									<m:semantics>
										<m:msup>
											<m:mi>a</m:mi>
											<m:mo>†</m:mo>
										</m:msup>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">superscript</m:csymbol>
												<m:ci>a</m:ci>
												<m:ci>†</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>)
and number (<m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>) operators are
defined in analogy with the conventional boson operators. The general
form postulated is</p>
							<table class="equation" id="S3.E19">


								<tr valign="baseline" class="equation" id="S3.E19">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mi>a</m:mi>
													<m:mo>|</m:mo>
													<m:mi>l</m:mi>
													<m:mo>&gt;</m:mo>
													<m:mo>=</m:mo>
													<m:msqrt>
														<m:mfenced open="[" close="]">
															<m:mi>l</m:mi>
														</m:mfenced>
													</m:msqrt>
													<m:mo>|</m:mo>
													<m:mi>l</m:mi>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
													<m:mo>&gt;</m:mo>
													<m:mi>a</m:mi>
													<m:msup>
														<m:mi/>
														<m:mo>†</m:mo>
													</m:msup>
													<m:mo>|</m:mo>
													<m:mi>l</m:mi>
													<m:mo>&gt;</m:mo>
													<m:mo>=</m:mo>
													<m:msqrt>
														<m:mfenced open="[" close="]">
															<m:mrow>
																<m:mi>l</m:mi>
																<m:mo>+</m:mo>
																<m:mn>1</m:mn>
															</m:mrow>
														</m:mfenced>
													</m:msqrt>
													<m:mo>|</m:mo>
													<m:mi>l</m:mi>
													<m:mo>+</m:mo>
													<m:mn>1</m:mn>
													<m:mo>&gt;</m:mo>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
													<m:mo>|</m:mo>
													<m:mi>l</m:mi>
													<m:mo>≥</m:mo>
													<m:mi>l</m:mi>
													<m:mo>|</m:mo>
													<m:mi>l</m:mi>
													<m:mo>&gt;</m:mo>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:ci>a</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">19</span>)</td>
								</tr>

							</table>
							<p class="p">It follows immediately that <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mo>†</m:mo>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:mi>a</m:mi>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mfenced open="[" close="]">
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
													<m:ci>a</m:ci>
												</m:apply>
												<m:apply>
													<m:ci>⌃</m:ci>
													<m:ci>n</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi>a</m:mi>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mo>†</m:mo>
												</m:msup>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mfenced open="[" close="]">
												<m:mrow>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
													<m:mo>+</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:ci>a</m:ci>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:plus/>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
													<m:cn>1</m:cn>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
The two widely used forms of the deformed
bosons are obtained by choosing either
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mfenced open="[" close="]">
												<m:mi>l</m:mi>
											</m:mfenced>
											<m:mo>=</m:mo>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>l</m:mi>
												</m:mfenced>
												<m:mi>M</m:mi>
											</m:msub>
											<m:mo>=</m:mo>
											<m:mfrac>
												<m:mrow>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mi>l</m:mi>
													</m:msup>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
												<m:mrow>
													<m:mi>q</m:mi>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:mfrac>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci/>
												<m:ci>l</m:ci>
												<m:eq/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>l</m:ci>
													<m:ci>M</m:ci>
												</m:apply>
												<m:eq/>
												<m:apply>
													<m:divide/>
													<m:apply>
														<m:minus/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:ci>l</m:ci>
														</m:apply>
														<m:cn>1</m:cn>
													</m:apply>
													<m:apply>
														<m:minus/>
														<m:ci>q</m:ci>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> or
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mfenced open="[" close="]">
												<m:mi>l</m:mi>
											</m:mfenced>
											<m:mo>=</m:mo>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>l</m:mi>
												</m:mfenced>
												<m:mi>P</m:mi>
											</m:msub>
											<m:mo>=</m:mo>
											<m:mfrac>
												<m:mrow>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mi>l</m:mi>
													</m:msup>
													<m:mo>-</m:mo>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mrow>
															<m:mo>-</m:mo>
															<m:mi>l</m:mi>
														</m:mrow>
													</m:msup>
												</m:mrow>
												<m:mrow>
													<m:mi>q</m:mi>
													<m:mo>-</m:mo>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mrow>
															<m:mo>-</m:mo>
															<m:mn>1</m:mn>
														</m:mrow>
													</m:msup>
												</m:mrow>
											</m:mfrac>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci/>
												<m:ci>l</m:ci>
												<m:eq/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>l</m:ci>
													<m:ci>P</m:ci>
												</m:apply>
												<m:eq/>
												<m:apply>
													<m:divide/>
													<m:apply>
														<m:minus/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:ci>l</m:ci>
														</m:apply>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:apply>
																<m:minus/>
																<m:ci>l</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:minus/>
														<m:ci>q</m:ci>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:apply>
																<m:minus/>
																<m:cn>1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
A generalisation was recently proposed by
Chakrabarti and Jagannathan <cite class="cite">[<a href="#bib.bib14" title="" class="ref">14</a>]</cite>. We shall adhere to
the notation introduced by Wachs and White <cite class="cite">[<a href="#bib.bib12" title="" class="ref">12</a>]</cite> and
write this generalisation in the form
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mfenced open="[" close="]">
												<m:mi>l</m:mi>
											</m:mfenced>
											<m:mo>=</m:mo>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>l</m:mi>
												</m:mfenced>
												<m:mi>G</m:mi>
											</m:msub>
											<m:mo>=</m:mo>
											<m:mfrac>
												<m:mrow>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mi>l</m:mi>
													</m:msup>
													<m:mo>-</m:mo>
													<m:msup>
														<m:mi>p</m:mi>
														<m:mi>l</m:mi>
													</m:msup>
												</m:mrow>
												<m:mrow>
													<m:mi>q</m:mi>
													<m:mo>-</m:mo>
													<m:mi>p</m:mi>
												</m:mrow>
											</m:mfrac>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci/>
												<m:ci>l</m:ci>
												<m:eq/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>l</m:ci>
													<m:ci>G</m:ci>
												</m:apply>
												<m:eq/>
												<m:apply>
													<m:divide/>
													<m:apply>
														<m:minus/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:ci>l</m:ci>
														</m:apply>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>p</m:ci>
															<m:ci>l</m:ci>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:minus/>
														<m:ci>q</m:ci>
														<m:ci>p</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, which is
trivially modified relative to that introduced in
Ref. <cite class="cite">[<a href="#bib.bib14" title="" class="ref">14</a>]</cite>.
As a consequence of a remark made in the previous section, this
third type of deformed boson contains the first two as special cases.</p>
						</div>

						<div class="para" id="S3.p2">
							<p class="p">The deformed bosons as defined by Eq. (<a href="#S3.E19" title="Eq.19 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">19</a>) are not associated
with any a priori specification of a (possibly deformed)
commutation relation. Choosing a parameter <m:math display="inline">
									<m:semantics>
										<m:mi>Q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>Q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, which does not have to
be related to the two parameters <m:math display="inline">
									<m:semantics>
										<m:mi>p</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>p</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math display="inline">
									<m:semantics>
										<m:mi>q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> so far introduced,
the deformed bosons are found to satisfy the deformed commutation
relation</p>
							<table class="equation" id="S3.E20">


								<tr valign="baseline" class="equation" id="S3.E20">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mi>a</m:mi>
																	<m:mo>,</m:mo>
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mrow>
															</m:mfenced>
															<m:mi>Q</m:mi>
														</m:msub>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mrow>
																<m:mi>a</m:mi>
																<m:mo>⁢</m:mo>
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mo>†</m:mo>
																</m:msup>
															</m:mrow>
															<m:mo>-</m:mo>
															<m:mrow>
																<m:mi>Q</m:mi>
																<m:mo>⁢</m:mo>
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mo>†</m:mo>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:mi>a</m:mi>
															</m:mrow>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mi>ϕ</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mfrac>
																<m:mn>1</m:mn>
																<m:mrow>
																	<m:mi>q</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>p</m:mi>
																</m:mrow>
															</m:mfrac>
															<m:mo>⁢</m:mo>
															<m:none/>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:msup>
																			<m:mi>q</m:mi>
																			<m:mover accent="true">
																				<m:mi>n</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																		</m:msup>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>q</m:mi>
																				<m:mo>-</m:mo>
																				<m:mi>Q</m:mi>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>+</m:mo>
																	<m:mrow>
																		<m:msup>
																			<m:mi>p</m:mi>
																			<m:mover accent="true">
																				<m:mi>n</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																		</m:msup>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>Q</m:mi>
																				<m:mo>-</m:mo>
																				<m:mi>p</m:mi>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo>⁢</m:mo>
																		<m:none/>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:apply>
																<m:interval closure="closed"/>
																<m:ci>a</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
															</m:apply>
															<m:ci>Q</m:ci>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:times/>
																<m:ci>a</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>Q</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>a</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>ϕ</m:ci>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>n</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:divide/>
																<m:cn>1</m:cn>
																<m:apply>
																	<m:minus/>
																	<m:ci>q</m:ci>
																	<m:ci>p</m:ci>
																</m:apply>
															</m:apply>
															<m:ci/>
															<m:apply>
																<m:plus/>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>q</m:ci>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:minus/>
																		<m:ci>q</m:ci>
																		<m:ci>Q</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>p</m:ci>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:minus/>
																		<m:ci>Q</m:ci>
																		<m:ci>p</m:ci>
																	</m:apply>
																	<m:ci/>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">20</span>)</td>
								</tr>

							</table>
							<p class="p">Since the choice of <m:math display="inline">
									<m:semantics>
										<m:mi>Q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>Q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is arbitrary we can opt to be guided by the
requirement that the form of <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>ϕ</m:mi>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:ci>ϕ</m:ci>
												<m:apply>
													<m:ci>⌃</m:ci>
													<m:ci>n</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> be as simple as possible
or by some other relevant criterion.
The conventional choice <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mi>q</m:mi>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:ci>q</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, to which we will eventually adhere,
results in</p>
							<table class="equation" id="S3.E21">


								<tr valign="baseline" class="equation" id="S3.E21">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mrow>
																<m:mi>a</m:mi>
																<m:mo>⁢</m:mo>
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mo>†</m:mo>
																</m:msup>
															</m:mrow>
															<m:mo>-</m:mo>
															<m:mrow>
																<m:mi>q</m:mi>
																<m:mo>⁢</m:mo>
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mo>†</m:mo>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:mi>a</m:mi>
															</m:mrow>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msub>
																<m:mi>ϕ</m:mi>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
													<m:mo>,</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:times/>
																<m:ci>a</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>a</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>ϕ</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>n</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:cn>1</m:cn>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">21</span>)</td>
								</tr>

							</table>
							<table class="equation" id="S3.E22">


								<tr valign="baseline" class="equation" id="S3.E22">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mi>a</m:mi>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mo>†</m:mo>
															</m:msup>
														</m:mrow>
														<m:mo>-</m:mo>
														<m:mrow>
															<m:mi>q</m:mi>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mo>†</m:mo>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:mi>a</m:mi>
														</m:mrow>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msub>
															<m:mi>ϕ</m:mi>
															<m:mi>P</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mover accent="true">
																<m:mi>n</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mrow>
															<m:mo>-</m:mo>
															<m:mover accent="true">
																<m:mi>n</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
														</m:mrow>
													</m:msup>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:times/>
																<m:ci>a</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>a</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>ϕ</m:ci>
																<m:ci>P</m:ci>
															</m:apply>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>n</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:apply>
																<m:minus/>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">22</span>)</td>
								</tr>

							</table>
							<p class="p">and</p>
							<table class="equation" id="S3.E23">


								<tr valign="baseline" class="equation" id="S3.E23">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mi>a</m:mi>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mo>†</m:mo>
															</m:msup>
														</m:mrow>
														<m:mo>-</m:mo>
														<m:mrow>
															<m:mi>q</m:mi>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mo>†</m:mo>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:mi>a</m:mi>
														</m:mrow>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msub>
															<m:mi>ϕ</m:mi>
															<m:mi>G</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mover accent="true">
																<m:mi>n</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:msup>
														<m:mi>p</m:mi>
														<m:mover accent="true">
															<m:mi>n</m:mi>
															<m:mo>⌃</m:mo>
														</m:mover>
													</m:msup>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:times/>
																<m:ci>a</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>a</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>ϕ</m:ci>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>n</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>p</m:ci>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>n</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">23</span>)</td>
								</tr>

							</table>
							<p class="p">for the M-type, P-type and G-type bosons, respectively. We do not
label the creation and annihilation operators by indices such
as M, P or G because the nature of these operators is always
obvious from the context.
The choice <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mi>p</m:mi>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:ci>p</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> results in
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi>ϕ</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:ci>ϕ</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> for all the three cases.
For the M-type bosons (<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>) this choice implies <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, i.e.,
the deformed commutation relation becomes
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mrow>
													<m:mi>a</m:mi>
													<m:mo>⁢</m:mo>
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
												</m:mrow>
												<m:mo>-</m:mo>
												<m:mrow>
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
													<m:mo>⁢</m:mo>
													<m:mi>a</m:mi>
												</m:mrow>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:minus/>
													<m:apply>
														<m:times/>
														<m:ci>a</m:ci>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:times/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
														<m:ci>a</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
For the P-type bosons (<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mrow>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:minus/>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>) this choice is the familiar
alternative to Eq. (<a href="#S3.E22" title="Eq.22 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">22</a>), namely
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mrow>
													<m:mi>a</m:mi>
													<m:mo>⁢</m:mo>
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
												</m:mrow>
												<m:mo>-</m:mo>
												<m:mrow>
													<m:msup>
														<m:mi>q</m:mi>
														<m:mrow>
															<m:mo>-</m:mo>
															<m:mn>1</m:mn>
														</m:mrow>
													</m:msup>
													<m:mo>⁢</m:mo>
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
													<m:mo>⁢</m:mo>
													<m:mi>a</m:mi>
												</m:mrow>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:minus/>
													<m:apply>
														<m:times/>
														<m:ci>a</m:ci>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:times/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:apply>
																<m:minus/>
																<m:cn>1</m:cn>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
														<m:ci>a</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
In a recent study of the extension of the
Campbell-Baker-Hausdorff formula to deformed bosons <cite class="cite">[<a href="#bib.bib16" title="" class="ref">16</a>]</cite>, it was
noted that the choice <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mi>q</m:mi>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:ci>q</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is the most suitable one for
the M-type bosons, but that <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mi>q</m:mi>
												<m:mo>+</m:mo>
												<m:msup>
													<m:mi>q</m:mi>
													<m:mrow>
														<m:mo>-</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
												</m:msup>
												<m:mo>-</m:mo>
												<m:mn>1</m:mn>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:apply>
													<m:minus/>
													<m:apply>
														<m:plus/>
														<m:ci>q</m:ci>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>q</m:ci>
															<m:apply>
																<m:minus/>
																<m:cn>1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:cn>1</m:cn>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> seems to have some advantages
for the P-type bosons. From the same point of view, one would
choose <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mi>q</m:mi>
												<m:mo>+</m:mo>
												<m:mi>p</m:mi>
												<m:mo>-</m:mo>
												<m:mn>1</m:mn>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:apply>
													<m:minus/>
													<m:apply>
														<m:plus/>
														<m:ci>q</m:ci>
														<m:ci>p</m:ci>
													</m:apply>
													<m:cn>1</m:cn>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> for the G-type bosons.</p>
						</div>

						<div class="para" id="S3.p3">
							<p class="p">We shall also need the relation</p>
							<table class="equation" id="S3.E24">


								<tr valign="baseline" class="equation" id="S3.E24">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msub>
														<m:mfenced open="[" close="]">
															<m:mrow>
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mi>k</m:mi>
																</m:msup>
																<m:mo>,</m:mo>
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mo>†</m:mo>
																</m:msup>
															</m:mrow>
														</m:mfenced>
														<m:msup>
															<m:mi>Q</m:mi>
															<m:mi>k</m:mi>
														</m:msup>
													</m:msub>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mi mathvariant="normal">Φ</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>a</m:mi>
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>-</m:mo>
																<m:mn>1</m:mn>
															</m:mrow>
														</m:msup>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:apply>
																<m:interval closure="closed"/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>Q</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:ci>Φ</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:ci>k</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">24</span>)</td>
								</tr>

							</table>
							<p class="p">which can be viewed as an extension of Eq. (<a href="#S3.E20" title="Eq.20 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">20</a>) in the sense
that <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi mathvariant="normal">Φ</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mn>1</m:mn>
														<m:mo>,</m:mo>
														<m:mover accent="true">
															<m:mi>n</m:mi>
															<m:mo>⌃</m:mo>
														</m:mover>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mi>ϕ</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:ci>Φ</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:cn>1</m:cn>
														<m:apply>
															<m:ci>⌃</m:ci>
															<m:ci>n</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:times/>
													<m:ci>ϕ</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
One easily finds that</p>
							<table class="equation" id="S3.E25">


								<tr valign="baseline" class="equation" id="S3.E25">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mi mathvariant="normal">Φ</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mfrac>
																<m:mn>1</m:mn>
																<m:mrow>
																	<m:mi>q</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>p</m:mi>
																</m:mrow>
															</m:mfrac>
															<m:mo>⁢</m:mo>
															<m:none/>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:msup>
																			<m:mi>q</m:mi>
																			<m:mover accent="true">
																				<m:mi>n</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																		</m:msup>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>q</m:mi>
																				<m:mo>-</m:mo>
																				<m:mi>Q</m:mi>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo>⁢</m:mo>
																		<m:msub>
																			<m:mfenced open="[" close="]">
																				<m:mi>k</m:mi>
																			</m:mfenced>
																			<m:mrow>
																				<m:mi>G</m:mi>
																				<m:mo>⁢</m:mo>
																				<m:mfenced open="(" close=")">
																					<m:mrow>
																						<m:mi>Q</m:mi>
																						<m:mo>,</m:mo>
																						<m:mi>q</m:mi>
																					</m:mrow>
																				</m:mfenced>
																			</m:mrow>
																		</m:msub>
																	</m:mrow>
																	<m:mo>+</m:mo>
																	<m:mrow>
																		<m:msup>
																			<m:mi>p</m:mi>
																			<m:mover accent="true">
																				<m:mi>n</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																		</m:msup>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>Q</m:mi>
																				<m:mo>-</m:mo>
																				<m:mi>p</m:mi>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo>⁢</m:mo>
																		<m:msub>
																			<m:mfenced open="[" close="]">
																				<m:mi>k</m:mi>
																			</m:mfenced>
																			<m:mrow>
																				<m:mi>G</m:mi>
																				<m:mo>⁢</m:mo>
																				<m:mfenced open="(" close=")">
																					<m:mrow>
																						<m:mi>Q</m:mi>
																						<m:mo>,</m:mo>
																						<m:mi>p</m:mi>
																					</m:mrow>
																				</m:mfenced>
																			</m:mrow>
																		</m:msub>
																		<m:mo>⁢</m:mo>
																		<m:none/>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>Φ</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:divide/>
																<m:cn>1</m:cn>
																<m:apply>
																	<m:minus/>
																	<m:ci>q</m:ci>
																	<m:ci>p</m:ci>
																</m:apply>
															</m:apply>
															<m:ci/>
															<m:apply>
																<m:plus/>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>q</m:ci>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:minus/>
																		<m:ci>q</m:ci>
																		<m:ci>Q</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>k</m:ci>
																		<m:apply>
																			<m:times/>
																			<m:ci>G</m:ci>
																			<m:apply>
																				<m:interval closure="open"/>
																				<m:ci>Q</m:ci>
																				<m:ci>q</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>p</m:ci>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:minus/>
																		<m:ci>Q</m:ci>
																		<m:ci>p</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>k</m:ci>
																		<m:apply>
																			<m:times/>
																			<m:ci>G</m:ci>
																			<m:apply>
																				<m:interval closure="open"/>
																				<m:ci>Q</m:ci>
																				<m:ci>p</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:ci/>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">25</span>)</td>
								</tr>

							</table>
							<p class="p">We shall retain the conventional choice <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mi>q</m:mi>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:ci>q</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
for the three cases specified above.
With this choice we get</p>
							<table class="equation" id="S3.E26">


								<tr valign="baseline" class="equation" id="S3.E26">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msub>
																<m:mi mathvariant="normal">Φ</m:mi>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>k</m:mi>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mi mathvariant="normal">Φ</m:mi>
																<m:mi>P</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>k</m:mi>
																</m:mfenced>
																<m:mi>P</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mrow>
																	<m:mo>-</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mi mathvariant="normal">Φ</m:mi>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>k</m:mi>
																</m:mfenced>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>p</m:mi>
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>Φ</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>k</m:ci>
																<m:ci>M</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>Φ</m:ci>
																<m:ci>P</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>k</m:ci>
																<m:ci>P</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>Φ</m:ci>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>k</m:ci>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>p</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">26</span>)</td>
								</tr>

							</table>
						</div>

					</div>

					<div class="section" id="S4">
						<h2 class="title section-title">
							<span class="refnum">
								<span class="reftype">§ </span>4. </span>Normal ordering of powers of the deformed number operator</h2>
						<div class="para" id="S4.p1">
							<p class="p">The relevance of the ordinary Stirling numbers to the normal ordering of
powers of the boson number operator was demonstrated in
Ref. <cite class="cite">[<a href="#bib.bib15" title="" class="ref">15</a>]</cite>.
In the present section we consider some normal ordering properties
of the deformed bosons specified by the parameter choice <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mi>q</m:mi>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:ci>q</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, which corresponds to the M-type boson operators
and to the deformed commutation relation (<a href="#S3.E21" title="Eq.21 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">21</a>).
Up to a trivial interchange of <m:math display="inline">
									<m:semantics>
										<m:mi>p</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>p</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math display="inline">
									<m:semantics>
										<m:mi>q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
this is the only combination of parameters for which the
deformed commutator does not depend on <m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
The other types of deformed boson operators are considered in the
following
section where it is found that they differ in a significant respect from
the case presently considered.</p>
						</div>

						<div class="para" id="S4.p2">
							<p class="p">In order to express an integral power of <m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:mfenced>
											<m:mi>M</m:mi>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:apply>
													<m:ci>⌃</m:ci>
													<m:ci>n</m:ci>
												</m:apply>
												<m:ci>M</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
in a normally ordered form we can either formally write
such an expansion and obtain a recurrence relation for
the coefficients by applying Eq. (<a href="#S3.E21" title="Eq.21 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">21</a>) or
use the deformed Stirling
numbers of the second kind directly. We shall present
both approaches because of the intrinsic interest of each one of them.</p>
						</div>

						<div class="para" id="S4.p3">
							<p class="p">In the direct approach, we start from the expansion</p>
							<table class="equation" id="S4.E27">


								<tr valign="baseline" class="equation" id="S4.E27">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mfenced>
															<m:mi>M</m:mi>
															<m:mi>m</m:mi>
														</m:msubsup>
														<m:mo>=</m:mo>
														<m:msup>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:mi>a</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mi>m</m:mi>
														</m:msup>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>m</m:mi>
															</m:mover>
															<m:mi>c</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>a</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>c</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>a</m:ci>
																		<m:ci>†</m:ci>
																	</m:apply>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">27</span>)</td>
								</tr>

							</table>
							<p class="p">Expressing <m:math display="inline">
									<m:semantics>
										<m:msup>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
													<m:mo>⁢</m:mo>
													<m:mi>a</m:mi>
												</m:mrow>
											</m:mfenced>
											<m:mrow>
												<m:mi>m</m:mi>
												<m:mo>+</m:mo>
												<m:mn>1</m:mn>
											</m:mrow>
										</m:msup>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">superscript</m:csymbol>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
													<m:ci>a</m:ci>
												</m:apply>
												<m:apply>
													<m:plus/>
													<m:ci>m</m:ci>
													<m:cn>1</m:cn>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> by means of Eq. (<a href="#S4.E27" title="Eq.27 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">27</a>) and
using Eq. (<a href="#S3.E21" title="Eq.21 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">21</a>), we obtain
a recurrence relation
which is identical with the one satisfied by <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mi>S</m:mi>
												<m:mi>q</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>S</m:ci>
													<m:ci>q</m:ci>
												</m:apply>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
Eq. (<a href="#S2.E9" title="Eq.9 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">9</a>). Moreover,
it is obvious from the defining equation (<a href="#S4.E27" title="Eq.27 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">27</a>) that
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi>c</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mn>1</m:mn>
														<m:mo>,</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:msub>
													<m:mi>S</m:mi>
													<m:mi>q</m:mi>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mn>1</m:mn>
														<m:mo>,</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci/>
												<m:apply>
													<m:times/>
													<m:ci>c</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:cn>1</m:cn>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>S</m:ci>
														<m:ci>q</m:ci>
													</m:apply>
													<m:apply>
														<m:interval closure="open"/>
														<m:cn>1</m:cn>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
												<m:eq/>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. Thus, <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi>c</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>m</m:mi>
														<m:mo>,</m:mo>
														<m:mi>k</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:msub>
													<m:mi>S</m:mi>
													<m:mi>q</m:mi>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>m</m:mi>
														<m:mo>,</m:mo>
														<m:mi>k</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:ci>c</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>m</m:ci>
														<m:ci>k</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>S</m:ci>
														<m:ci>q</m:ci>
													</m:apply>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>m</m:ci>
														<m:ci>k</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="para" id="S4.p4">
							<p class="p">A different derivation can be obtained by using the identity</p>
							<table class="equation" id="S4.E28">


								<tr valign="baseline" class="equation" id="S4.E28">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∏</m:mo>
																	<m:mrow>
																		<m:mi>i</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>0</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>-</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mover>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mrow>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																		<m:mo>-</m:mo>
																		<m:mi>i</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>∏</m:ci>
																	<m:apply>
																		<m:eq/>
																		<m:ci>i</m:ci>
																		<m:cn>0</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:minus/>
																	<m:ci>k</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:minus/>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																	<m:ci>i</m:ci>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">28</span>)</td>
								</tr>

							</table>
							<p class="p">This identity follows by noting that application of both sides of
Eq. (<a href="#S4.E28" title="Eq.28 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">28</a>) on any member of the complete set
								<m:math display="inline">
									<m:semantics>
										<m:mfenced open="{" close="}">
											<m:mrow>
												<m:mrow>
													<m:mrow>
														<m:mfenced open="|" close="&gt;">
															<m:mi>l</m:mi>
														</m:mfenced>
														<m:mo>;</m:mo>
														<m:mi>l</m:mi>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mn>0</m:mn>
												</m:mrow>
												<m:mo>,</m:mo>
												<m:mrow>
													<m:mn> 1</m:mn>
													<m:mo>,</m:mo>
													<m:mi mathvariant="normal">⋯</m:mi>
												</m:mrow>
											</m:mrow>
										</m:mfenced>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:set/>
												<m:apply>
													<m:ci/>
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:list/>
															<m:apply>
																<m:ci/>
																<m:ci>l</m:ci>
															</m:apply>
															<m:ci>l</m:ci>
														</m:apply>
														<m:cn>0</m:cn>
													</m:apply>
													<m:apply>
														<m:list/>
														<m:cn> 1</m:cn>
														<m:ci>⋯</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
of eigenstates of the number operator
results in <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msubsup>
												<m:mo>∏</m:mo>
												<m:mrow>
													<m:mi>i</m:mi>
													<m:mo>=</m:mo>
													<m:mn>0</m:mn>
												</m:mrow>
												<m:mrow>
													<m:mi>k</m:mi>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mrow>
														<m:mi>l</m:mi>
														<m:mo>-</m:mo>
														<m:mi>i</m:mi>
													</m:mrow>
												</m:mfenced>
												<m:mi>M</m:mi>
											</m:msub>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>∏</m:ci>
														<m:apply>
															<m:eq/>
															<m:ci>i</m:ci>
															<m:cn>0</m:cn>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:minus/>
														<m:ci>k</m:ci>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:minus/>
														<m:ci>l</m:ci>
														<m:ci>i</m:ci>
													</m:apply>
													<m:ci>M</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
Using Eq. (<a href="#S2.E7" title="Eq.7 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">7</a>) we obtain</p>
							<table class="equation" id="S4.E29">


								<tr valign="baseline" class="equation" id="S4.E29">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msubsup>
														<m:mfenced open="[" close="]">
															<m:mover accent="true">
																<m:mi>n</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
														</m:mfenced>
														<m:mi>M</m:mi>
														<m:mi>m</m:mi>
													</m:msubsup>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mover>
															<m:munder>
																<m:mo movablelimits="false">∑</m:mo>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo movablelimits="false">=</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:munder>
															<m:mi>m</m:mi>
														</m:mover>
														<m:msub>
															<m:mi>S</m:mi>
															<m:mi>q</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>m</m:mi>
																<m:mo>,</m:mo>
																<m:mi>k</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∏</m:mo>
																	<m:mrow>
																		<m:mi>i</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>0</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>-</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mover>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mrow>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																		<m:mo>-</m:mo>
																		<m:mi>i</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
														</m:mrow>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:apply>
																			<m:csymbol cd="ambiguous">subscript</m:csymbol>
																			<m:ci>∏</m:ci>
																			<m:apply>
																				<m:eq/>
																				<m:ci>i</m:ci>
																				<m:cn>0</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:apply>
																			<m:minus/>
																			<m:ci>k</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:apply>
																			<m:minus/>
																			<m:apply>
																				<m:ci>⌃</m:ci>
																				<m:ci>n</m:ci>
																			</m:apply>
																			<m:ci>i</m:ci>
																		</m:apply>
																		<m:ci>M</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">29</span>)</td>
								</tr>

							</table>
							<p class="p">and substituting Eq. (<a href="#S4.E28" title="Eq.28 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">28</a>) we get the desired normally ordered
expansion</p>
							<table class="equation" id="S4.E30">


								<tr valign="baseline" class="equation" id="S4.E30">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mfenced>
															<m:mi>M</m:mi>
															<m:mi>m</m:mi>
														</m:msubsup>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>m</m:mi>
															</m:mover>
															<m:msub>
																<m:mi>S</m:mi>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>S</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>a</m:ci>
																		<m:ci>†</m:ci>
																	</m:apply>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">30</span>)</td>
								</tr>

							</table>
							<p class="p">We note in passing that an equivalent expansion could have been obtained
starting from the identity</p>
							<table class="equation" id="S4.E31">


								<tr valign="baseline" class="equation" id="S4.E31">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∏</m:mo>
																	<m:mrow>
																		<m:mi>i</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>0</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>-</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mover>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mover accent="true">
																				<m:mi>n</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																		</m:mfenced>
																		<m:mi>M</m:mi>
																	</m:msub>
																	<m:mo>-</m:mo>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mi>i</m:mi>
																		</m:mfenced>
																		<m:mi>M</m:mi>
																	</m:msub>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mrow>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>k</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>/</m:mo>
																	<m:mn>2</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>∏</m:ci>
																	<m:apply>
																		<m:eq/>
																		<m:ci>i</m:ci>
																		<m:cn>0</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:minus/>
																	<m:ci>k</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:minus/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																	<m:ci>M</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>i</m:ci>
																	<m:ci>M</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:divide/>
																	<m:apply>
																		<m:times/>
																		<m:ci>k</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>k</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:cn>2</m:cn>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">31</span>)</td>
								</tr>

							</table>
							<p class="p">This identity can be proved either
by induction or by considering the effect
of both sides on the complete set of eigenstates of the number operator.
Using (<a href="#S2.E11" title="Eq.11 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">11</a>) and (<a href="#S4.E31" title="Eq.31 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">31</a>), we obtain the normally ordered
expansion of <m:math display="inline">
									<m:semantics>
										<m:msubsup>
											<m:mfenced open="[" close="]">
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:mfenced>
											<m:mi>M</m:mi>
											<m:mi>m</m:mi>
										</m:msubsup>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">superscript</m:csymbol>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
													<m:ci>M</m:ci>
												</m:apply>
												<m:ci>m</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
in the form</p>
							<table class="equation" id="S4.E32">


								<tr valign="baseline" class="equation" id="S4.E32">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msubsup>
														<m:mfenced open="[" close="]">
															<m:mover accent="true">
																<m:mi>n</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
														</m:mfenced>
														<m:mi>M</m:mi>
														<m:mi>m</m:mi>
													</m:msubsup>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mover>
															<m:munder>
																<m:mo movablelimits="false">∑</m:mo>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo movablelimits="false">=</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:munder>
															<m:mi>m</m:mi>
														</m:mover>
														<m:msub>
															<m:mover accent="true">
																<m:mi>S</m:mi>
																<m:mo>~</m:mo>
															</m:mover>
															<m:mi>q</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>m</m:mi>
																<m:mo>,</m:mo>
																<m:mi>k</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>q</m:mi>
															<m:mrow>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>⁢</m:mo>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
																<m:mo>/</m:mo>
																<m:mn>2</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mfenced open="(" close=")">
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mo>†</m:mo>
																</m:msup>
															</m:mfenced>
															<m:mi>k</m:mi>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>a</m:mi>
															<m:mi>k</m:mi>
														</m:msup>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>~</m:ci>
																		<m:ci>S</m:ci>
																	</m:apply>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:ci>k</m:ci>
																			<m:apply>
																				<m:minus/>
																				<m:ci>k</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>a</m:ci>
																		<m:ci>†</m:ci>
																	</m:apply>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">32</span>)</td>
								</tr>

							</table>
							<p class="p">which is related to (<a href="#S4.E30" title="Eq.30 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">30</a>) by Eq. (<a href="#S2.E15" title="Eq.15 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">15</a>).</p>
						</div>

						<div class="para" id="S4.p5">
							<p class="p">In order  to obtain the inverse relation,
expressing a normally ordered product
as a function of the number operator, we note that
Eqs. (<a href="#S2.E6" title="Eq.6 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">6</a>) and (<a href="#S4.E28" title="Eq.28 in §4. Normal ordering of powers of the deformed number operator in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">28</a>) lead to</p>
							<table class="equation" id="S4.E33">


								<tr valign="baseline" class="equation" id="S4.E33">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mrow>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mi mathvariant="normal">⋯</m:mi>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mrow>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																		<m:mo>-</m:mo>
																		<m:mi>k</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:mfenced>
																<m:mi>M</m:mi>
															</m:msub>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>k</m:mi>
															</m:mover>
															<m:msub>
																<m:mi>s</m:mi>
																<m:mi>q</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>m</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msubsup>
																<m:mfenced open="[" close="]">
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mfenced>
																<m:mi>M</m:mi>
																<m:mi>m</m:mi>
															</m:msubsup>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:minus/>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
															<m:ci>⋯</m:ci>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:plus/>
																	<m:apply>
																		<m:minus/>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																		<m:ci>k</m:ci>
																	</m:apply>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>M</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>s</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																		<m:ci>M</m:ci>
																	</m:apply>
																	<m:ci>m</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">33</span>)</td>
								</tr>

							</table>
						</div>

					</div>

					<div class="section" id="S5">
						<h2 class="title section-title">
							<span class="refnum">
								<span class="reftype">§ </span>5. </span>Operator-valued deformed Stirling numbers</h2>
						<div class="para" id="S5.p1">
							<p class="p">In the present section, we attempt to derive
the normally ordered expansion of a power of the number operator for
arbitrarily deformed bosons.
Allowing <m:math display="inline">
									<m:semantics>
										<m:mi>p</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>p</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, <m:math display="inline">
									<m:semantics>
										<m:mi>q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math display="inline">
									<m:semantics>
										<m:mi>Q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>Q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> to be arbitrary,
we demand</p>
							<table class="equation" id="S5.E34">


								<tr valign="baseline" class="equation" id="S5.E34">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mfenced>
															<m:mi>G</m:mi>
															<m:mi>m</m:mi>
														</m:msubsup>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>m</m:mi>
															</m:mover>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:mover accent="true">
																<m:mi>S</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
																<m:ci>G</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>a</m:ci>
																		<m:ci>†</m:ci>
																	</m:apply>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:apply>
																	<m:vector/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">34</span>)</td>
								</tr>

							</table>
							<p class="p">Using the general relation (<a href="#S3.E24" title="Eq.24 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">24</a>),
we derive the recurrence relation</p>
							<table class="equation" id="S5.E35">


								<tr valign="baseline" class="equation" id="S5.E35">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mover accent="true">
																<m:mi>S</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mrow>
																<m:msup>
																	<m:mi>Q</m:mi>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																		<m:mo>,</m:mo>
																		<m:mrow>
																			<m:mover accent="true">
																				<m:mi>n</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																			<m:mo>+</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo>+</m:mo>
															<m:mrow>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>k</m:mi>
																		<m:mo>,</m:mo>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																	</m:mrow>
																</m:mfenced>
																<m:mo>⁢</m:mo>
																<m:mi mathvariant="normal">Φ</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>,</m:mo>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>S</m:ci>
															</m:apply>
															<m:apply>
																<m:vector/>
																<m:apply>
																	<m:plus/>
																	<m:ci>m</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>Q</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:apply>
																	<m:vector/>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																	<m:apply>
																		<m:plus/>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:apply>
																	<m:vector/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
																<m:ci>Φ</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">35</span>)</td>
								</tr>

							</table>
							<p class="p">The “boundary conditions” and initial values, for all values of
								<m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, are the same as those following Eq. (<a href="#S2.E4" title="Eq.4 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">4</a>).</p>
						</div>

						<div class="para" id="S5.p2">
							<p class="p">The M-type bosons (<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>), with the choice <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mi>q</m:mi>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:ci>q</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> which yields
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:msub>
													<m:mi mathvariant="normal">Φ</m:mi>
													<m:mi>M</m:mi>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>k</m:mi>
														<m:mo>,</m:mo>
														<m:mover accent="true">
															<m:mi>n</m:mi>
															<m:mo>⌃</m:mo>
														</m:mover>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mi>k</m:mi>
												</m:mfenced>
												<m:mi>M</m:mi>
											</m:msub>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>Φ</m:ci>
														<m:ci>M</m:ci>
													</m:apply>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>k</m:ci>
														<m:apply>
															<m:ci>⌃</m:ci>
															<m:ci>n</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>k</m:ci>
													<m:ci>M</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, were studied in section 4.
For this case, <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mover accent="true">
												<m:mi>S</m:mi>
												<m:mo>⌃</m:mo>
											</m:mover>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:ci>⌃</m:ci>
													<m:ci>S</m:ci>
												</m:apply>
												<m:apply>
													<m:vector/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> does not depend on
								<m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. More specifically,
Eq. (<a href="#S5.E35" title="Eq.35 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">35</a>) then reduces
to Eq. (<a href="#S2.E9" title="Eq.9 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">9</a>). For the
G-type bosons, we found in section 3 that by choosing <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mi>q</m:mi>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:ci>q</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> we
obtain <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:msub>
													<m:mi mathvariant="normal">Φ</m:mi>
													<m:mi>G</m:mi>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>k</m:mi>
														<m:mo>,</m:mo>
														<m:mover accent="true">
															<m:mi>n</m:mi>
															<m:mo>⌃</m:mo>
														</m:mover>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:msub>
													<m:mfenced open="[" close="]">
														<m:mi>k</m:mi>
													</m:mfenced>
													<m:mi>G</m:mi>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>p</m:mi>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:msup>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>Φ</m:ci>
														<m:ci>G</m:ci>
													</m:apply>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>k</m:ci>
														<m:apply>
															<m:ci>⌃</m:ci>
															<m:ci>n</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>k</m:ci>
														<m:ci>G</m:ci>
													</m:apply>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>p</m:ci>
														<m:apply>
															<m:ci>⌃</m:ci>
															<m:ci>n</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> ; consequently,
we have</p>
							<table class="equation" id="S5.E36">


								<tr valign="baseline" class="equation" id="S5.E36">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mrow>
																<m:msup>
																	<m:mi>q</m:mi>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mover accent="true">
																		<m:mi>S</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																	<m:mi>G</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																		<m:mo>,</m:mo>
																		<m:mrow>
																			<m:mover accent="true">
																				<m:mi>n</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																			<m:mo>+</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo>+</m:mo>
															<m:mrow>
																<m:msub>
																	<m:mover accent="true">
																		<m:mi>S</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																	<m:mi>G</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>k</m:mi>
																		<m:mo>,</m:mo>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																	</m:mrow>
																</m:mfenced>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>k</m:mi>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:msup>
																	<m:mi>p</m:mi>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:msup>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:vector/>
																<m:apply>
																	<m:plus/>
																	<m:ci>m</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>S</m:ci>
																	</m:apply>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:apply>
																	<m:vector/>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																	<m:apply>
																		<m:plus/>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>S</m:ci>
																	</m:apply>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:apply>
																	<m:vector/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>k</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">36</span>)</td>
								</tr>

							</table>
							<p class="p">Note that in the general case
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mover accent="true">
													<m:mi>S</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
												<m:mi>G</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>S</m:ci>
													</m:apply>
													<m:ci>G</m:ci>
												</m:apply>
												<m:apply>
													<m:vector/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> depends on the operator <m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
The special cases <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mrow>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:minus/>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> are contained in
Eq. (<a href="#S5.E36" title="Eq.36 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">36</a>).
The dependence of <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mover accent="true">
													<m:mi>S</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
												<m:mi>G</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>S</m:ci>
													</m:apply>
													<m:ci>G</m:ci>
												</m:apply>
												<m:apply>
													<m:vector/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> on <m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> for all cases
except <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> can be taken to imply that we have
actually failed to obtain a normally
ordered expansion for <m:math display="inline">
									<m:semantics>
										<m:msubsup>
											<m:mfenced open="[" close="]">
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:mfenced>
											<m:mi>G</m:mi>
											<m:mi>m</m:mi>
										</m:msubsup>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">superscript</m:csymbol>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
													<m:ci>G</m:ci>
												</m:apply>
												<m:ci>m</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> in terms of a finite sum in
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msup>
												<m:mfenced open="(" close=")">
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
												</m:mfenced>
												<m:mi>k</m:mi>
											</m:msup>
											<m:mo>⁢</m:mo>
											<m:msup>
												<m:mi>a</m:mi>
												<m:mi>k</m:mi>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
													<m:ci>k</m:ci>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>a</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> with <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>k</m:mi>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mn>1</m:mn>
												<m:mo>,</m:mo>
												<m:mn> 2</m:mn>
												<m:mo>,</m:mo>
												<m:mi mathvariant="normal">⋯</m:mi>
												<m:mo>,</m:mo>
												<m:mi>m</m:mi>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>k</m:ci>
												<m:apply>
													<m:list/>
													<m:cn>1</m:cn>
													<m:cn> 2</m:cn>
													<m:ci>⋯</m:ci>
													<m:ci>m</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="para" id="S5.p3">
							<p class="p">The structure of the recurrence  relation (<a href="#S5.E36" title="Eq.36 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">36</a>) indicates
that the dependence on <m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> of the deformed Stirling
numbers <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mover accent="true">
													<m:mi>S</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
												<m:mi>G</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>S</m:ci>
													</m:apply>
													<m:ci>G</m:ci>
												</m:apply>
												<m:apply>
													<m:vector/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
can be expressed in terms of the factor
								<m:math display="inline">
									<m:semantics>
										<m:msup>
											<m:mi>p</m:mi>
											<m:mrow>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>m</m:mi>
														<m:mo>-</m:mo>
														<m:mi>k</m:mi>
													</m:mrow>
												</m:mfenced>
												<m:mo>⁢</m:mo>
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:mrow>
										</m:msup>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">superscript</m:csymbol>
												<m:ci>p</m:ci>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:minus/>
														<m:ci>m</m:ci>
														<m:ci>k</m:ci>
													</m:apply>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. Defining the (<m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-independent) reduced
Stirling numbers of the second kind
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi mathvariant="normal">Ξ</m:mi>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:ci>Ξ</m:ci>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> through</p>
							<table class="equation" id="S5.E37">


								<tr valign="baseline" class="equation" id="S5.E37">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mover accent="true">
																<m:mi>S</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
															<m:mi>G</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>m</m:mi>
																<m:mo>,</m:mo>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msup>
															<m:mi>q</m:mi>
															<m:mrow>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>⁢</m:mo>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
																<m:mo>/</m:mo>
																<m:mn>2</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>p</m:mi>
															<m:mrow>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo>-</m:mo>
																		<m:mi>k</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:mo>⁢</m:mo>
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:mi mathvariant="normal">Ξ</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>m</m:mi>
																<m:mo>,</m:mo>
																<m:mi>k</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:vector/>
																<m:ci>m</m:ci>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:divide/>
																	<m:apply>
																		<m:times/>
																		<m:ci>k</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>k</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:cn>2</m:cn>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>p</m:ci>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:minus/>
																		<m:ci>m</m:ci>
																		<m:ci>k</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:ci>Ξ</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>m</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">37</span>)</td>
								</tr>

							</table>
							<p class="p">we obtain the recurrence relation</p>
							<table class="equation" id="S5.E38">


								<tr valign="baseline" class="equation" id="S5.E38">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mi mathvariant="normal">Ξ</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>+</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
																<m:mo>,</m:mo>
																<m:mi>k</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mrow>
															<m:msup>
																<m:mi>p</m:mi>
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>+</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:mi mathvariant="normal">Ξ</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>-</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>+</m:mo>
														<m:mrow>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>k</m:mi>
																</m:mfenced>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mi mathvariant="normal">Ξ</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>Ξ</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>m</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:plus/>
																		<m:apply>
																			<m:minus/>
																			<m:ci>m</m:ci>
																			<m:ci>k</m:ci>
																		</m:apply>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>Ξ</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>k</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:ci>Ξ</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">38</span>)</td>
								</tr>

							</table>
							<p class="p">with  the initial condition <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi mathvariant="normal">Ξ</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mn>1</m:mn>
														<m:mo>,</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:ci>Ξ</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:cn>1</m:cn>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="para" id="S5.p4">
							<p class="p">To obtain
the “inverse relation” to (<a href="#S5.E34" title="Eq.34 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">34</a>),
expressing a normally ordered term
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msup>
												<m:mfenced open="(" close=")">
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
												</m:mfenced>
												<m:mi>k</m:mi>
											</m:msup>
											<m:mo>⁢</m:mo>
											<m:msup>
												<m:mi>a</m:mi>
												<m:mi>k</m:mi>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
													<m:ci>k</m:ci>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>a</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> by means of a polynomial in <m:math display="inline">
									<m:semantics>
										<m:msub>
											<m:mfenced open="[" close="]">
												<m:mover accent="true">
													<m:mi>n</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
											</m:mfenced>
											<m:mi>G</m:mi>
										</m:msub>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">subscript</m:csymbol>
												<m:apply>
													<m:ci>⌃</m:ci>
													<m:ci>n</m:ci>
												</m:apply>
												<m:ci>G</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
we need the “G-arithmetic” identity</p>
							<table class="equation" id="S5.E39">


								<tr valign="baseline" class="equation" id="S5.E39">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mi>a</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>b</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mi>G</m:mi>
														</m:msub>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mrow>
																	<m:mo>-</m:mo>
																	<m:mi>b</m:mi>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mi>a</m:mi>
																		</m:mfenced>
																		<m:mi>G</m:mi>
																	</m:msub>
																	<m:mo>-</m:mo>
																	<m:mrow>
																		<m:msup>
																			<m:mi>p</m:mi>
																			<m:mrow>
																				<m:mi>a</m:mi>
																				<m:mo>-</m:mo>
																				<m:mi>b</m:mi>
																			</m:mrow>
																		</m:msup>
																		<m:mo>⁢</m:mo>
																		<m:msub>
																			<m:mfenced open="[" close="]">
																				<m:mi>b</m:mi>
																			</m:mfenced>
																			<m:mi>G</m:mi>
																		</m:msub>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>,</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:apply>
																<m:minus/>
																<m:ci>a</m:ci>
																<m:ci>b</m:ci>
															</m:apply>
															<m:ci>G</m:ci>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:ci>b</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:minus/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>p</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>a</m:ci>
																			<m:ci>b</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>b</m:ci>
																		<m:ci>G</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">39</span>)</td>
								</tr>

							</table>
							<p class="p">which follows from the two identities</p>
							<table class="equation" id="S5.E40">


								<tr valign="baseline" class="equation" id="S5.E40">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msub>
														<m:mfenced open="[" close="]">
															<m:mrow>
																<m:mi>a</m:mi>
																<m:mo>+</m:mo>
																<m:mi>b</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mi>G</m:mi>
													</m:msub>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mrow>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mi>b</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>a</m:mi>
																</m:mfenced>
																<m:mi>G</m:mi>
															</m:msub>
														</m:mrow>
														<m:mo>+</m:mo>
														<m:mrow>
															<m:msup>
																<m:mi>p</m:mi>
																<m:mi>a</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>b</m:mi>
																</m:mfenced>
																<m:mi>G</m:mi>
															</m:msub>
														</m:mrow>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:apply>
																<m:plus/>
																<m:ci>a</m:ci>
																<m:ci>b</m:ci>
															</m:apply>
															<m:ci>G</m:ci>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:ci>b</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:ci>a</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>b</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">40</span>)</td>
								</tr>

							</table>
							<p class="p">and</p>
							<table class="equation" id="S5.E41">


								<tr valign="baseline" class="equation" id="S5.E41">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mo>-</m:mo>
																	<m:mi>b</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mi>G</m:mi>
														</m:msub>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mo>-</m:mo>
															<m:mrow>
																<m:msup>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>p</m:mi>
																			<m:mo>⁢</m:mo>
																			<m:mi>q</m:mi>
																		</m:mrow>
																	</m:mfenced>
																	<m:mrow>
																		<m:mo>-</m:mo>
																		<m:mi>b</m:mi>
																	</m:mrow>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>b</m:mi>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:apply>
																<m:minus/>
																<m:ci>b</m:ci>
															</m:apply>
															<m:ci>G</m:ci>
														</m:apply>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:times/>
																		<m:ci>p</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:minus/>
																		<m:ci>b</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>b</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">41</span>)</td>
								</tr>

							</table>
							<p class="p">We now proceed to obtain the desired relation</p>
							<table class="equation" id="S5.E42">


								<tr valign="baseline" class="equation" id="S5.E42">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>m</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>k</m:mi>
															</m:mover>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>s</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
															<m:mo>⁢</m:mo>
															<m:msubsup>
																<m:mfenced open="[" close="]">
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mfenced>
																<m:mi>G</m:mi>
																<m:mi>m</m:mi>
															</m:msubsup>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>s</m:ci>
																	</m:apply>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:apply>
																	<m:vector/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																		<m:ci>G</m:ci>
																	</m:apply>
																	<m:ci>m</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">42</span>)</td>
								</tr>

							</table>
							<p class="p">Since  <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:msup>
													<m:mfenced open="(" close=")">
														<m:msup>
															<m:mi>a</m:mi>
															<m:mo>†</m:mo>
														</m:msup>
													</m:mfenced>
													<m:mrow>
														<m:mi>k</m:mi>
														<m:mo>+</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mrow>
														<m:mi>k</m:mi>
														<m:mo>+</m:mo>
														<m:mn>1</m:mn>
													</m:mrow>
												</m:msup>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:msup>
													<m:mfenced open="(" close=")">
														<m:msup>
															<m:mi>a</m:mi>
															<m:mo>†</m:mo>
														</m:msup>
													</m:mfenced>
													<m:mi>k</m:mi>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:msub>
													<m:mfenced open="[" close="]">
														<m:mover accent="true">
															<m:mi>n</m:mi>
															<m:mo>⌃</m:mo>
														</m:mover>
													</m:mfenced>
													<m:mi>G</m:mi>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mi>k</m:mi>
												</m:msup>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:msup>
													<m:mfenced open="(" close=")">
														<m:msup>
															<m:mi>a</m:mi>
															<m:mo>†</m:mo>
														</m:msup>
													</m:mfenced>
													<m:mi>k</m:mi>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:msup>
													<m:mi>a</m:mi>
													<m:mi>k</m:mi>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:msub>
													<m:mfenced open="[" close="]">
														<m:mrow>
															<m:mover accent="true">
																<m:mi>n</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
															<m:mo>-</m:mo>
															<m:mi>k</m:mi>
														</m:mrow>
													</m:mfenced>
													<m:mi>G</m:mi>
												</m:msub>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci/>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:ci>k</m:ci>
															<m:cn>1</m:cn>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:apply>
															<m:plus/>
															<m:ci>k</m:ci>
															<m:cn>1</m:cn>
														</m:apply>
													</m:apply>
												</m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
														<m:ci>k</m:ci>
													</m:apply>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:apply>
															<m:ci>⌃</m:ci>
															<m:ci>n</m:ci>
														</m:apply>
														<m:ci>G</m:ci>
													</m:apply>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>k</m:ci>
													</m:apply>
												</m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
														<m:ci>k</m:ci>
													</m:apply>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>k</m:ci>
													</m:apply>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>n</m:ci>
															</m:apply>
															<m:ci>k</m:ci>
														</m:apply>
														<m:ci>G</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
we can use Eqs. (<a href="#S5.E39" title="Eq.39 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">39</a>) and (<a href="#S5.E42" title="Eq.42 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">42</a>) to obtain
the recurrence relation</p>
							<table class="equation" id="S5.E43">


								<tr valign="baseline" class="equation" id="S5.E43">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>s</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																	<m:mo>,</m:mo>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mrow>
																	<m:mo>-</m:mo>
																	<m:mi>k</m:mi>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:none/>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:msub>
																			<m:mover accent="true">
																				<m:mi>s</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																			<m:mi>G</m:mi>
																		</m:msub>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>k</m:mi>
																				<m:mo>,</m:mo>
																				<m:mrow>
																					<m:mi>m</m:mi>
																					<m:mo>-</m:mo>
																					<m:mn>1</m:mn>
																				</m:mrow>
																				<m:mo>,</m:mo>
																				<m:mover accent="true">
																					<m:mi>n</m:mi>
																					<m:mo>⌃</m:mo>
																				</m:mover>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>-</m:mo>
																	<m:mrow>
																		<m:msup>
																			<m:mi>p</m:mi>
																			<m:mrow>
																				<m:mover accent="true">
																					<m:mi>n</m:mi>
																					<m:mo>⌃</m:mo>
																				</m:mover>
																				<m:mo>-</m:mo>
																				<m:mi>k</m:mi>
																			</m:mrow>
																		</m:msup>
																		<m:mo>⁢</m:mo>
																		<m:msub>
																			<m:mfenced open="[" close="]">
																				<m:mi>k</m:mi>
																			</m:mfenced>
																			<m:mi>G</m:mi>
																		</m:msub>
																		<m:mo>⁢</m:mo>
																		<m:msub>
																			<m:mover accent="true">
																				<m:mi>s</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																			<m:mi>G</m:mi>
																		</m:msub>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>k</m:mi>
																				<m:mo>,</m:mo>
																				<m:mi>m</m:mi>
																				<m:mo>,</m:mo>
																				<m:mover accent="true">
																					<m:mi>n</m:mi>
																					<m:mo>⌃</m:mo>
																				</m:mover>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo>⁢</m:mo>
																		<m:none/>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>s</m:ci>
																</m:apply>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:vector/>
																<m:apply>
																	<m:plus/>
																	<m:ci>k</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>m</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
															<m:ci/>
															<m:apply>
																<m:minus/>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>s</m:ci>
																		</m:apply>
																		<m:ci>G</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:vector/>
																		<m:ci>k</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>m</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>p</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:apply>
																				<m:ci>⌃</m:ci>
																				<m:ci>n</m:ci>
																			</m:apply>
																			<m:ci>k</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>k</m:ci>
																		<m:ci>G</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>s</m:ci>
																		</m:apply>
																		<m:ci>G</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:vector/>
																		<m:ci>k</m:ci>
																		<m:ci>m</m:ci>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:ci/>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">43</span>)</td>
								</tr>

							</table>
							<p class="p">Note that for <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> this recurrence relation reduces to
Eq. (<a href="#S2.E8" title="Eq.8 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">8</a>).</p>
						</div>

						<div class="para" id="S5.p5">
							<p class="p">Introducing the (<m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-independent) reduced Stirling numbers
of the first kind
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>ξ</m:mi>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mi>m</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:ci>ξ</m:ci>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>k</m:ci>
													<m:ci>m</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> such that</p>
							<table class="equation" id="S5.E44">


								<tr valign="baseline" class="equation" id="S5.E44">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mover accent="true">
																<m:mi>s</m:mi>
																<m:mo>⌃</m:mo>
															</m:mover>
															<m:mi>G</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
																<m:mo>,</m:mo>
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msup>
															<m:mi>q</m:mi>
															<m:mrow>
																<m:mo>-</m:mo>
																<m:mrow>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>k</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>/</m:mo>
																	<m:mn>2</m:mn>
																</m:mrow>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>p</m:mi>
															<m:mrow>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>-</m:mo>
																		<m:mi>m</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:mo>⁢</m:mo>
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:mi>ξ</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>,</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>s</m:ci>
																</m:apply>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:vector/>
																<m:ci>k</m:ci>
																<m:ci>m</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>q</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:ci>k</m:ci>
																			<m:apply>
																				<m:minus/>
																				<m:ci>k</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>p</m:ci>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:ci>m</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:ci>ξ</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>k</m:ci>
																<m:ci>m</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">44</span>)</td>
								</tr>

							</table>
							<p class="p">in Eq. (<a href="#S5.E43" title="Eq.43 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">43</a>), we obtain the recurrence relation</p>
							<table class="equation" id="S5.E45">


								<tr valign="baseline" class="equation" id="S5.E45">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mi>ξ</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																	<m:mo>,</m:mo>
																	<m:mi>m</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mrow>
																<m:mi>ξ</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>,</m:mo>
																		<m:mrow>
																			<m:mi>m</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo>-</m:mo>
															<m:mrow>
																<m:msup>
																	<m:mi>p</m:mi>
																	<m:mrow>
																		<m:mo>-</m:mo>
																		<m:mi>k</m:mi>
																	</m:mrow>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>k</m:mi>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
																<m:mo>⁢</m:mo>
																<m:mi>ξ</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>m</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>ξ</m:ci>
															<m:apply>
																<m:interval closure="open"/>
																<m:apply>
																	<m:plus/>
																	<m:ci>k</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:times/>
																<m:ci>ξ</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>m</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>k</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:ci>ξ</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">45</span>)</td>
								</tr>

							</table>
						</div>

						<div class="para" id="S5.p6">
							<p class="p">The exponential dependence on <m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> of the
deformed Stirling numbers of the first kind, <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mover accent="true">
													<m:mi>s</m:mi>
													<m:mo>⌃</m:mo>
												</m:mover>
												<m:mi>G</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>s</m:ci>
													</m:apply>
													<m:ci>G</m:ci>
												</m:apply>
												<m:apply>
													<m:vector/>
													<m:ci>k</m:ci>
													<m:ci>m</m:ci>
													<m:apply>
														<m:ci>⌃</m:ci>
														<m:ci>n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
means that we have not been able to
express <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msup>
												<m:mfenced open="(" close=")">
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
												</m:mfenced>
												<m:mi>k</m:mi>
											</m:msup>
											<m:mo>⁢</m:mo>
											<m:msup>
												<m:mi>a</m:mi>
												<m:mi>k</m:mi>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
													<m:ci>k</m:ci>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>a</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> as
a <span style="" class="text italic">polynomial</span> in <m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> but
we did express it as a
								<span style="" class="text italic">function</span> of <m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="para" id="S5.p7">
							<p class="p">In order to derive the bi-orthogonality relations between the
deformed Stirling numbers of the first and second kinds, we first
rewrite Eq. (<a href="#S5.E34" title="Eq.34 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">34</a>) in the form</p>
							<table class="equation" id="S5.E46">


								<tr valign="baseline" class="equation" id="S5.E46">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mfenced>
															<m:mi>G</m:mi>
															<m:mi>m</m:mi>
														</m:msubsup>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>1</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>m</m:mi>
															</m:mover>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mrow>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																		<m:mo>-</m:mo>
																		<m:mi>k</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
																<m:ci>G</m:ci>
															</m:apply>
															<m:ci>m</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>a</m:ci>
																		<m:ci>†</m:ci>
																	</m:apply>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>S</m:ci>
																	</m:apply>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:apply>
																	<m:vector/>
																	<m:ci>m</m:ci>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																		<m:ci>k</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">46</span>)</td>
								</tr>

							</table>
							<p class="p">Using Eq. (<a href="#S5.E37" title="Eq.37 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">37</a>) we obtain</p>
							<table class="equation" id="S5.E47">


								<tr valign="baseline" class="equation" id="S5.E47">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mrow>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																		<m:mo>-</m:mo>
																		<m:mi>k</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:msup>
																<m:mi>p</m:mi>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>⁢</m:mo>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mi>m</m:mi>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mover accent="true">
																	<m:mi>S</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>m</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>k</m:mi>
																	<m:mo>,</m:mo>
																	<m:mover accent="true">
																		<m:mi>n</m:mi>
																		<m:mo>⌃</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:vector/>
																<m:ci>m</m:ci>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:minus/>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																	<m:ci>k</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>p</m:ci>
																<m:apply>
																	<m:times/>
																	<m:ci>k</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>k</m:ci>
																		<m:ci>m</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>S</m:ci>
																</m:apply>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:vector/>
																<m:ci>m</m:ci>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:ci>⌃</m:ci>
																	<m:ci>n</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">47</span>)</td>
								</tr>

							</table>
							<p class="p">Defining <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:msup>
													<m:mi mathvariant="normal">Ξ</m:mi>
													<m:mo>′</m:mo>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>m</m:mi>
														<m:mo>,</m:mo>
														<m:mi>k</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:msup>
													<m:mi>p</m:mi>
													<m:mrow>
														<m:mi>k</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>-</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:msup>
												<m:mo>⁢</m:mo>
												<m:mi mathvariant="normal">Ξ</m:mi>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mi>m</m:mi>
														<m:mo>,</m:mo>
														<m:mi>k</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>Ξ</m:ci>
														<m:ci>′</m:ci>
													</m:apply>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>m</m:ci>
														<m:ci>k</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>p</m:ci>
														<m:apply>
															<m:times/>
															<m:ci>k</m:ci>
															<m:apply>
																<m:minus/>
																<m:ci>k</m:ci>
																<m:ci>m</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:ci>Ξ</m:ci>
													<m:apply>
														<m:interval closure="open"/>
														<m:ci>m</m:ci>
														<m:ci>k</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
we obtain relations of the form of Eqs. (<a href="#S2.E16" title="Eq.16 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">16</a>) and (<a href="#S2.E17" title="Eq.17 in §2. Stirling and deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">17</a>)
with <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msup>
												<m:mi mathvariant="normal">Ξ</m:mi>
												<m:mo>′</m:mo>
											</m:msup>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>Ξ</m:ci>
													<m:ci>′</m:ci>
												</m:apply>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> replacing <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mi>S</m:mi>
												<m:mi>q</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>m</m:mi>
													<m:mo>,</m:mo>
													<m:mi>k</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>S</m:ci>
													<m:ci>q</m:ci>
												</m:apply>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>m</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
and <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>ξ</m:mi>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mi>m</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:ci>ξ</m:ci>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>k</m:ci>
													<m:ci>m</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> replacing <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mi>s</m:mi>
												<m:mi>q</m:mi>
											</m:msub>
											<m:mo>⁢</m:mo>
											<m:mfenced open="(" close=")">
												<m:mrow>
													<m:mi>k</m:mi>
													<m:mo>,</m:mo>
													<m:mi>m</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:ci>s</m:ci>
													<m:ci>q</m:ci>
												</m:apply>
												<m:apply>
													<m:interval closure="open"/>
													<m:ci>k</m:ci>
													<m:ci>m</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
							</p>
						</div>

					</div>

					<div class="section" id="S6">
						<h2 class="title section-title">
							<span class="refnum">
								<span class="reftype">§ </span>6. </span>A generating function for the deformed Stirling numbers
of the first kind</h2>
						<div class="para" id="S6.p1">
							<p class="p">We start by transforming the <m:math display="inline">
									<m:semantics>
										<m:mi>q</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>q</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-binomial theorem <cite class="cite">[<a href="#bib.bib13" title="" class="ref">13</a>]</cite> into a
G-binomial theorem. By introducing the symbol</p>
							<table class="equation" id="S6.E48">


								<tr valign="baseline" class="equation" id="S6.E48">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msup>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>λ</m:mi>
																<m:mo>;</m:mo>
																<m:mi>x</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mfenced open="(" close=")">
															<m:mi>l</m:mi>
														</m:mfenced>
													</m:msup>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mi>λ</m:mi>
																<m:mo>+</m:mo>
																<m:mi>x</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:mi>p</m:mi>
																	<m:mo>⁢</m:mo>
																	<m:mi>λ</m:mi>
																</m:mrow>
																<m:mo>+</m:mo>
																<m:mrow>
																	<m:mi>q</m:mi>
																	<m:mo>⁢</m:mo>
																	<m:mi>x</m:mi>
																</m:mrow>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:msup>
																		<m:mi>p</m:mi>
																		<m:mn>2</m:mn>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:mi>λ</m:mi>
																</m:mrow>
																<m:mo>+</m:mo>
																<m:mrow>
																	<m:msup>
																		<m:mi>q</m:mi>
																		<m:mn>2</m:mn>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:mi>x</m:mi>
																</m:mrow>
															</m:mrow>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:mi mathvariant="normal">⋯</m:mi>
														<m:mo>⁢</m:mo>
														<m:mfenced open="(" close=")">
															<m:mrow>
																<m:mrow>
																	<m:msup>
																		<m:mi>p</m:mi>
																		<m:mrow>
																			<m:mi>l</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:mi>λ</m:mi>
																</m:mrow>
																<m:mo>+</m:mo>
																<m:mrow>
																	<m:msup>
																		<m:mi>q</m:mi>
																		<m:mrow>
																			<m:mi>l</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:mi>x</m:mi>
																</m:mrow>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>λ</m:ci>
																<m:ci>x</m:ci>
															</m:apply>
															<m:ci>l</m:ci>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:plus/>
																<m:ci>λ</m:ci>
																<m:ci>x</m:ci>
															</m:apply>
															<m:apply>
																<m:plus/>
																<m:apply>
																	<m:times/>
																	<m:ci>p</m:ci>
																	<m:ci>λ</m:ci>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:ci>q</m:ci>
																	<m:ci>x</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:plus/>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>p</m:ci>
																		<m:cn>2</m:cn>
																	</m:apply>
																	<m:ci>λ</m:ci>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>q</m:ci>
																		<m:cn>2</m:cn>
																	</m:apply>
																	<m:ci>x</m:ci>
																</m:apply>
															</m:apply>
															<m:ci>⋯</m:ci>
															<m:apply>
																<m:plus/>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>p</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>l</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:ci>λ</m:ci>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>q</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>l</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:ci>x</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">48</span>)</td>
								</tr>

							</table>
							<p class="p">we have</p>
							<table class="equation" id="S6.E49">


								<tr valign="baseline" class="equation" id="S6.E49">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msup>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>λ</m:mi>
																	<m:mo>;</m:mo>
																	<m:mi>x</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mfenced open="(" close=")">
																<m:mi>l</m:mi>
															</m:mfenced>
														</m:msup>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>i</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>0</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi>l</m:mi>
															</m:mover>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mtable rowspacing="0.2ex" columnspacing="0.4em">
																		<m:mtr>
																			<m:mtd/>
																			<m:mtd columnalign="center">
																				<m:mi>l</m:mi>
																			</m:mtd>
																			<m:mtd/>
																		</m:mtr>
																		<m:mtr>
																			<m:mtd/>
																			<m:mtd columnalign="center">
																				<m:mi>i</m:mi>
																			</m:mtd>
																			<m:mtd/>
																		</m:mtr>
																	</m:mtable>
																</m:mfenced>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>p</m:mi>
																<m:mrow>
																	<m:mrow>
																		<m:mi>i</m:mi>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>i</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>/</m:mo>
																	<m:mn>2</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mrow>
																	<m:mrow>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>l</m:mi>
																				<m:mo>-</m:mo>
																				<m:mi>i</m:mi>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>l</m:mi>
																				<m:mo>-</m:mo>
																				<m:mi>i</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>/</m:mo>
																	<m:mn>2</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>λ</m:mi>
																<m:mi>i</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>x</m:mi>
																<m:mrow>
																	<m:mi>l</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>i</m:mi>
																</m:mrow>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:mo>,</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>λ</m:ci>
																<m:ci>x</m:ci>
															</m:apply>
															<m:ci>l</m:ci>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>i</m:ci>
																		<m:cn>0</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>l</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:mtext>li</m:mtext>
																	<m:ci>G</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:ci>i</m:ci>
																			<m:apply>
																				<m:minus/>
																				<m:ci>i</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:apply>
																				<m:minus/>
																				<m:ci>l</m:ci>
																				<m:ci>i</m:ci>
																			</m:apply>
																			<m:apply>
																				<m:minus/>
																				<m:ci>l</m:ci>
																				<m:ci>i</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>λ</m:ci>
																	<m:ci>i</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>x</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:ci>l</m:ci>
																		<m:ci>i</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">49</span>)</td>
								</tr>

							</table>
							<p class="p">where</p>
							<table class="equation" id="S6.E50">


								<tr valign="baseline" class="equation" id="S6.E50">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msub>
														<m:mfenced open="[" close="]">
															<m:mtable rowspacing="0.2ex" columnspacing="0.4em">
																<m:mtr>
																	<m:mtd/>
																	<m:mtd columnalign="center">
																		<m:mi>l</m:mi>
																	</m:mtd>
																	<m:mtd/>
																</m:mtr>
																<m:mtr>
																	<m:mtd/>
																	<m:mtd columnalign="center">
																		<m:mi>i</m:mi>
																	</m:mtd>
																	<m:mtd/>
																</m:mtr>
															</m:mtable>
														</m:mfenced>
														<m:mi>G</m:mi>
													</m:msub>
													<m:mo>=</m:mo>
													<m:mfrac>
														<m:mrow>
															<m:msub>
																<m:mfenced open="[" close="]">
																	<m:mi>l</m:mi>
																</m:mfenced>
																<m:mi>G</m:mi>
															</m:msub>
															<m:mi mathvariant="normal">!</m:mi>
														</m:mrow>
														<m:mrow>
															<m:mrow>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>i</m:mi>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
																<m:mi mathvariant="normal">!</m:mi>
															</m:mrow>
															<m:mo>⁢</m:mo>
															<m:mrow>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mrow>
																			<m:mi>l</m:mi>
																			<m:mo>-</m:mo>
																			<m:mi>i</m:mi>
																		</m:mrow>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
																<m:mi mathvariant="normal">!</m:mi>
															</m:mrow>
														</m:mrow>
													</m:mfrac>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:mtext>li</m:mtext>
															<m:ci>G</m:ci>
														</m:apply>
														<m:apply>
															<m:divide/>
															<m:apply>
																<m:factorial/>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:ci>l</m:ci>
																	<m:ci>G</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:factorial/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>i</m:ci>
																		<m:ci>G</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:factorial/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:apply>
																			<m:minus/>
																			<m:ci>l</m:ci>
																			<m:ci>i</m:ci>
																		</m:apply>
																		<m:ci>G</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">50</span>)</td>
								</tr>

							</table>
							<p class="p">is a G-binomial coefficient and
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:msub>
													<m:mfenced open="[" close="]">
														<m:mi>k</m:mi>
													</m:mfenced>
													<m:mi>G</m:mi>
												</m:msub>
												<m:mi mathvariant="normal">!</m:mi>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:msub>
													<m:mfenced open="[" close="]">
														<m:mn>1</m:mn>
													</m:mfenced>
													<m:mi>G</m:mi>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:msub>
													<m:mfenced open="[" close="]">
														<m:mn>2</m:mn>
													</m:mfenced>
													<m:mi>G</m:mi>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:mi mathvariant="normal">⋯</m:mi>
												<m:mo>⁢</m:mo>
												<m:msub>
													<m:mfenced open="[" close="]">
														<m:mi>k</m:mi>
													</m:mfenced>
													<m:mi>G</m:mi>
												</m:msub>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:factorial/>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>k</m:ci>
														<m:ci>G</m:ci>
													</m:apply>
												</m:apply>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:cn>1</m:cn>
														<m:ci>G</m:ci>
													</m:apply>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:cn>2</m:cn>
														<m:ci>G</m:ci>
													</m:apply>
													<m:ci>⋯</m:ci>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>k</m:ci>
														<m:ci>G</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
Equation (<a href="#S6.E49" title="Eq.49 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">49</a>) can be proved by induction, using the
G-binomial coefficient recurrence relation</p>
							<table class="equation" id="S6.E51">


								<tr valign="baseline" class="equation" id="S6.E51">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mtable rowspacing="0.2ex" columnspacing="0.4em">
																	<m:mtr>
																		<m:mtd columnalign="center">
																			<m:mi>l</m:mi>
																		</m:mtd>
																		<m:mtd columnalign="center">
																			<m:mo>+</m:mo>
																		</m:mtd>
																		<m:mtd columnalign="center">
																			<m:mn>1</m:mn>
																		</m:mtd>
																	</m:mtr>
																	<m:mtr>
																		<m:mtd columnalign="center">
																			<m:none/>
																		</m:mtd>
																		<m:mtd columnalign="center">
																			<m:mi>i</m:mi>
																		</m:mtd>
																		<m:mtd columnalign="center">
																			<m:none/>
																		</m:mtd>
																	</m:mtr>
																</m:mtable>
															</m:mfenced>
															<m:mi>G</m:mi>
														</m:msub>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mrow>
																<m:msup>
																	<m:mi>p</m:mi>
																	<m:mrow>
																		<m:mi>l</m:mi>
																		<m:mo>+</m:mo>
																		<m:mn>1</m:mn>
																		<m:mo>-</m:mo>
																		<m:mi>i</m:mi>
																	</m:mrow>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mtable rowspacing="0.2ex" columnspacing="0.4em">
																			<m:mtr>
																				<m:mtd columnalign="center">
																					<m:none/>
																				</m:mtd>
																				<m:mtd columnalign="center">
																					<m:mi>l</m:mi>
																				</m:mtd>
																				<m:mtd columnalign="center">
																					<m:none/>
																				</m:mtd>
																			</m:mtr>
																			<m:mtr>
																				<m:mtd columnalign="center">
																					<m:mi>i</m:mi>
																				</m:mtd>
																				<m:mtd columnalign="center">
																					<m:mo>-</m:mo>
																				</m:mtd>
																				<m:mtd columnalign="center">
																					<m:mn>1</m:mn>
																				</m:mtd>
																			</m:mtr>
																		</m:mtable>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
															</m:mrow>
															<m:mo>+</m:mo>
															<m:mrow>
																<m:msup>
																	<m:mi>q</m:mi>
																	<m:mi>i</m:mi>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mtable rowspacing="0.2ex" columnspacing="0.4em">
																			<m:mtr>
																				<m:mtd/>
																				<m:mtd columnalign="center">
																					<m:mi>l</m:mi>
																				</m:mtd>
																				<m:mtd/>
																			</m:mtr>
																			<m:mtr>
																				<m:mtd/>
																				<m:mtd columnalign="center">
																					<m:mi>i</m:mi>
																				</m:mtd>
																				<m:mtd/>
																			</m:mtr>
																		</m:mtable>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo>,</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:mtext>l+1i</m:mtext>
															<m:ci>G</m:ci>
														</m:apply>
														<m:apply>
															<m:plus/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:apply>
																			<m:plus/>
																			<m:ci>l</m:ci>
																			<m:cn>1</m:cn>
																		</m:apply>
																		<m:ci>i</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:mtext>li-1</m:mtext>
																	<m:ci>G</m:ci>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:ci>i</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:mtext>li</m:mtext>
																	<m:ci>G</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">51</span>)</td>
								</tr>

							</table>
							<p class="p">which follows from the definition of the G-binomial coefficient
on using the G-arithmetic relation (<a href="#S5.E40" title="Eq.40 in §5. Operator-valued deformed Stirling numbers in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">40</a>).</p>
						</div>

						<div class="para" id="S6.p2">
							<p class="p">Now, from the identity</p>
							<table class="equation" id="S6.E52">


								<tr valign="baseline" class="equation" id="S6.E52">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mfrac>
															<m:mrow>
																<m:msup>
																	<m:mfenced open="(" close=")">
																		<m:msup>
																			<m:mi>a</m:mi>
																			<m:mo>†</m:mo>
																		</m:msup>
																	</m:mfenced>
																	<m:mi>k</m:mi>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mi>k</m:mi>
																</m:msup>
															</m:mrow>
															<m:mrow>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>k</m:mi>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
																<m:mi mathvariant="normal">!</m:mi>
															</m:mrow>
														</m:mfrac>
														<m:mo>⁢</m:mo>
														<m:mfenced open="|" close="&gt;">
															<m:mi>l</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mtable rowspacing="0.2ex" columnspacing="0.4em">
																	<m:mtr>
																		<m:mtd/>
																		<m:mtd columnalign="center">
																			<m:mi>l</m:mi>
																		</m:mtd>
																		<m:mtd/>
																	</m:mtr>
																	<m:mtr>
																		<m:mtd/>
																		<m:mtd columnalign="center">
																			<m:mi>k</m:mi>
																		</m:mtd>
																		<m:mtd/>
																	</m:mtr>
																</m:mtable>
															</m:mfenced>
															<m:mi>G</m:mi>
														</m:msub>
														<m:mo>⁢</m:mo>
														<m:mfenced open="|" close="&gt;">
															<m:mi>l</m:mi>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:divide/>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:apply>
																			<m:csymbol cd="ambiguous">superscript</m:csymbol>
																			<m:ci>a</m:ci>
																			<m:ci>†</m:ci>
																		</m:apply>
																		<m:ci>k</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>a</m:ci>
																		<m:ci>k</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:factorial/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>k</m:ci>
																		<m:ci>G</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:ci/>
																<m:ci>l</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:mtext>lk</m:mtext>
																<m:ci>G</m:ci>
															</m:apply>
															<m:apply>
																<m:ci/>
																<m:ci>l</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">52</span>)</td>
								</tr>

							</table>
							<p class="p">we obtain</p>
							<table class="equation" id="S6.E53">


								<tr valign="baseline" class="equation" id="S6.E53">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mover>
															<m:munder>
																<m:mo movablelimits="false">∑</m:mo>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo movablelimits="false">=</m:mo>
																	<m:mn>0</m:mn>
																</m:mrow>
															</m:munder>
															<m:mi>m</m:mi>
														</m:mover>
														<m:msup>
															<m:mi>p</m:mi>
															<m:mrow>
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>⁢</m:mo>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
																<m:mo>/</m:mo>
																<m:mn>2</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>q</m:mi>
															<m:mrow>
																<m:mrow>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>l</m:mi>
																			<m:mo>-</m:mo>
																			<m:mi>k</m:mi>
																		</m:mrow>
																	</m:mfenced>
																	<m:mo>⁢</m:mo>
																	<m:mfenced open="(" close=")">
																		<m:mrow>
																			<m:mi>l</m:mi>
																			<m:mo>-</m:mo>
																			<m:mi>k</m:mi>
																			<m:mo>-</m:mo>
																			<m:mn>1</m:mn>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
																<m:mo>/</m:mo>
																<m:mn>2</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>λ</m:mi>
															<m:mi>k</m:mi>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:mfrac>
															<m:mrow>
																<m:msup>
																	<m:mfenced open="(" close=")">
																		<m:msup>
																			<m:mi>a</m:mi>
																			<m:mo>†</m:mo>
																		</m:msup>
																	</m:mfenced>
																	<m:mi>k</m:mi>
																</m:msup>
																<m:mo>⁢</m:mo>
																<m:msup>
																	<m:mi>a</m:mi>
																	<m:mi>k</m:mi>
																</m:msup>
															</m:mrow>
															<m:mrow>
																<m:msub>
																	<m:mfenced open="[" close="]">
																		<m:mi>k</m:mi>
																	</m:mfenced>
																	<m:mi>G</m:mi>
																</m:msub>
																<m:mi mathvariant="normal">!</m:mi>
															</m:mrow>
														</m:mfrac>
														<m:mo>⁢</m:mo>
														<m:mfenced open="|" close="&gt;">
															<m:mi>l</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:msup>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>λ</m:mi>
																	<m:mo>;</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mfenced>
															<m:mfenced open="(" close=")">
																<m:mi>l</m:mi>
															</m:mfenced>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:mfenced open="|" close="&gt;">
															<m:mi>l</m:mi>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>0</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:ci>k</m:ci>
																			<m:apply>
																				<m:minus/>
																				<m:ci>k</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:apply>
																				<m:minus/>
																				<m:ci>l</m:ci>
																				<m:ci>k</m:ci>
																			</m:apply>
																			<m:apply>
																				<m:minus/>
																				<m:ci>l</m:ci>
																				<m:ci>k</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>λ</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:divide/>
																	<m:apply>
																		<m:times/>
																		<m:apply>
																			<m:csymbol cd="ambiguous">superscript</m:csymbol>
																			<m:apply>
																				<m:csymbol cd="ambiguous">superscript</m:csymbol>
																				<m:ci>a</m:ci>
																				<m:ci>†</m:ci>
																			</m:apply>
																			<m:ci>k</m:ci>
																		</m:apply>
																		<m:apply>
																			<m:csymbol cd="ambiguous">superscript</m:csymbol>
																			<m:ci>a</m:ci>
																			<m:ci>k</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:factorial/>
																		<m:apply>
																			<m:csymbol cd="ambiguous">subscript</m:csymbol>
																			<m:ci>k</m:ci>
																			<m:ci>G</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:ci/>
																	<m:ci>l</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>λ</m:ci>
																	<m:cn>1</m:cn>
																</m:apply>
																<m:ci>l</m:ci>
															</m:apply>
															<m:apply>
																<m:ci/>
																<m:ci>l</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">53</span>)</td>
								</tr>

							</table>
							<p class="p">which can be written as an operator identity</p>
							<table class="equation" id="S6.E54">


								<tr valign="baseline" class="equation" id="S6.E54">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:mover>
																<m:munder>
																	<m:mo movablelimits="false">∑</m:mo>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo movablelimits="false">=</m:mo>
																		<m:mn>0</m:mn>
																	</m:mrow>
																</m:munder>
																<m:mi mathvariant="normal">∞</m:mi>
															</m:mover>
															<m:msup>
																<m:mi>p</m:mi>
																<m:mrow>
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>k</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>/</m:mo>
																	<m:mn>2</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>q</m:mi>
																<m:mrow>
																	<m:mrow>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mover accent="true">
																					<m:mi>n</m:mi>
																					<m:mo>⌃</m:mo>
																				</m:mover>
																				<m:mo>-</m:mo>
																				<m:mi>k</m:mi>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mover accent="true">
																					<m:mi>n</m:mi>
																					<m:mo>⌃</m:mo>
																				</m:mover>
																				<m:mo>-</m:mo>
																				<m:mi>k</m:mi>
																				<m:mo>-</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>/</m:mo>
																	<m:mn>2</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>λ</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:mfrac>
																<m:mrow>
																	<m:msup>
																		<m:mfenced open="(" close=")">
																			<m:msup>
																				<m:mi>a</m:mi>
																				<m:mo>†</m:mo>
																			</m:msup>
																		</m:mfenced>
																		<m:mi>k</m:mi>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mi>k</m:mi>
																	</m:msup>
																</m:mrow>
																<m:mrow>
																	<m:msub>
																		<m:mfenced open="[" close="]">
																			<m:mi>k</m:mi>
																		</m:mfenced>
																		<m:mi>G</m:mi>
																	</m:msub>
																	<m:mi mathvariant="normal">!</m:mi>
																</m:mrow>
															</m:mfrac>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:msup>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>λ</m:mi>
																	<m:mo>;</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:mfenced>
															<m:mfenced open="(" close=")">
																<m:mover accent="true">
																	<m:mi>n</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mfenced>
														</m:msup>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>k</m:ci>
																		<m:cn>0</m:cn>
																	</m:apply>
																</m:apply>
																<m:infinity/>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:ci>k</m:ci>
																			<m:apply>
																				<m:minus/>
																				<m:ci>k</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:times/>
																			<m:apply>
																				<m:minus/>
																				<m:apply>
																					<m:ci>⌃</m:ci>
																					<m:ci>n</m:ci>
																				</m:apply>
																				<m:ci>k</m:ci>
																			</m:apply>
																			<m:apply>
																				<m:minus/>
																				<m:apply>
																					<m:ci>⌃</m:ci>
																					<m:ci>n</m:ci>
																				</m:apply>
																				<m:ci>k</m:ci>
																				<m:cn>1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>λ</m:ci>
																	<m:ci>k</m:ci>
																</m:apply>
																<m:apply>
																	<m:divide/>
																	<m:apply>
																		<m:times/>
																		<m:apply>
																			<m:csymbol cd="ambiguous">superscript</m:csymbol>
																			<m:apply>
																				<m:csymbol cd="ambiguous">superscript</m:csymbol>
																				<m:ci>a</m:ci>
																				<m:ci>†</m:ci>
																			</m:apply>
																			<m:ci>k</m:ci>
																		</m:apply>
																		<m:apply>
																			<m:csymbol cd="ambiguous">superscript</m:csymbol>
																			<m:ci>a</m:ci>
																			<m:ci>k</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:factorial/>
																		<m:apply>
																			<m:csymbol cd="ambiguous">subscript</m:csymbol>
																			<m:ci>k</m:ci>
																			<m:ci>G</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:apply>
																<m:interval closure="open"/>
																<m:ci>λ</m:ci>
																<m:cn>1</m:cn>
															</m:apply>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>n</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">54</span>)</td>
								</tr>

							</table>
							<p class="p">To obtain an expression for <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msup>
												<m:mfenced open="(" close=")">
													<m:msup>
														<m:mi>a</m:mi>
														<m:mo>†</m:mo>
													</m:msup>
												</m:mfenced>
												<m:mi>k</m:mi>
											</m:msup>
											<m:mo>⁢</m:mo>
											<m:msup>
												<m:mi>a</m:mi>
												<m:mi>k</m:mi>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:times/>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:apply>
														<m:csymbol cd="ambiguous">superscript</m:csymbol>
														<m:ci>a</m:ci>
														<m:ci>†</m:ci>
													</m:apply>
													<m:ci>k</m:ci>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>a</m:ci>
													<m:ci>k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> as a function of
the number operator <m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, we have to expand the right-hand side
of Eq. (<a href="#S6.E54" title="Eq.54 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">54</a>) in powers of <m:math display="inline">
									<m:semantics>
										<m:mi>λ</m:mi>
										<m:annotation-xml encoding="MathML-Content">
											<m:ci>λ</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
The coefficient of <m:math display="inline">
									<m:semantics>
										<m:msup>
											<m:mi>λ</m:mi>
											<m:mi>k</m:mi>
										</m:msup>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:csymbol cd="ambiguous">superscript</m:csymbol>
												<m:ci>λ</m:ci>
												<m:ci>k</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> can be extracted by writing
							</p>
							<table class="equation" id="S6.E55">


								<tr valign="baseline" class="equation" id="S6.E55">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mrow>
															<m:msup>
																<m:mfenced open="(" close=")">
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:mi>k</m:mi>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mi>k</m:mi>
															</m:msup>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:msub>
															<m:mfenced open="" close="|">
																<m:mrow>
																	<m:mfrac>
																		<m:mrow>
																			<m:msub>
																				<m:mfenced open="[" close="]">
																					<m:mi>k</m:mi>
																				</m:mfenced>
																				<m:mi>G</m:mi>
																			</m:msub>
																			<m:mi mathvariant="normal">!</m:mi>
																		</m:mrow>
																		<m:mrow>
																			<m:mi>k</m:mi>
																			<m:mi mathvariant="normal">!</m:mi>
																		</m:mrow>
																	</m:mfrac>
																	<m:mo>⁢</m:mo>
																	<m:msup>
																		<m:mi>p</m:mi>
																		<m:mrow>
																			<m:mo>-</m:mo>
																			<m:mrow>
																				<m:mrow>
																					<m:mi>k</m:mi>
																					<m:mo>⁢</m:mo>
																					<m:mfenced open="(" close=")">
																						<m:mrow>
																							<m:mi>k</m:mi>
																							<m:mo>-</m:mo>
																							<m:mn>1</m:mn>
																						</m:mrow>
																					</m:mfenced>
																				</m:mrow>
																				<m:mo>/</m:mo>
																				<m:mn>2</m:mn>
																			</m:mrow>
																		</m:mrow>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:msup>
																		<m:mi>q</m:mi>
																		<m:mrow>
																			<m:mo>-</m:mo>
																			<m:mrow>
																				<m:mrow>
																					<m:mfenced open="(" close=")">
																						<m:mrow>
																							<m:mover accent="true">
																								<m:mi>n</m:mi>
																								<m:mo>⌃</m:mo>
																							</m:mover>
																							<m:mo>-</m:mo>
																							<m:mi>k</m:mi>
																						</m:mrow>
																					</m:mfenced>
																					<m:mo>⁢</m:mo>
																					<m:mfenced open="(" close=")">
																						<m:mrow>
																							<m:mover accent="true">
																								<m:mi>n</m:mi>
																								<m:mo>⌃</m:mo>
																							</m:mover>
																							<m:mo>-</m:mo>
																							<m:mi>k</m:mi>
																							<m:mo>-</m:mo>
																							<m:mn>1</m:mn>
																						</m:mrow>
																					</m:mfenced>
																				</m:mrow>
																				<m:mo>/</m:mo>
																				<m:mn>2</m:mn>
																			</m:mrow>
																		</m:mrow>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:mfrac>
																		<m:msup>
																			<m:mo>∂</m:mo>
																			<m:mi>k</m:mi>
																		</m:msup>
																		<m:mrow>
																			<m:mo>∂</m:mo>
																			<m:mo>⁡</m:mo>
																			<m:msup>
																				<m:mi>λ</m:mi>
																				<m:mi>k</m:mi>
																			</m:msup>
																		</m:mrow>
																	</m:mfrac>
																	<m:mo>⁢</m:mo>
																	<m:msup>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>λ</m:mi>
																				<m:mo>;</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																		<m:mfenced open="(" close=")">
																			<m:mover accent="true">
																				<m:mi>n</m:mi>
																				<m:mo>⌃</m:mo>
																			</m:mover>
																		</m:mfenced>
																	</m:msup>
																	<m:mo>⁢</m:mo>
																	<m:none/>
																</m:mrow>
															</m:mfenced>
															<m:mrow>
																<m:mi>λ</m:mi>
																<m:mo>=</m:mo>
																<m:mn>0</m:mn>
															</m:mrow>
														</m:msub>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:ci/>
															<m:apply>
																<m:times/>
																<m:apply>
																	<m:divide/>
																	<m:apply>
																		<m:factorial/>
																		<m:apply>
																			<m:csymbol cd="ambiguous">subscript</m:csymbol>
																			<m:ci>k</m:ci>
																			<m:ci>G</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:factorial/>
																		<m:ci>k</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:apply>
																			<m:divide/>
																			<m:apply>
																				<m:times/>
																				<m:ci>k</m:ci>
																				<m:apply>
																					<m:minus/>
																					<m:ci>k</m:ci>
																					<m:cn>1</m:cn>
																				</m:apply>
																			</m:apply>
																			<m:cn>2</m:cn>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>q</m:ci>
																	<m:apply>
																		<m:minus/>
																		<m:apply>
																			<m:divide/>
																			<m:apply>
																				<m:times/>
																				<m:apply>
																					<m:minus/>
																					<m:apply>
																						<m:ci>⌃</m:ci>
																						<m:ci>n</m:ci>
																					</m:apply>
																					<m:ci>k</m:ci>
																				</m:apply>
																				<m:apply>
																					<m:minus/>
																					<m:apply>
																						<m:ci>⌃</m:ci>
																						<m:ci>n</m:ci>
																					</m:apply>
																					<m:ci>k</m:ci>
																					<m:cn>1</m:cn>
																				</m:apply>
																			</m:apply>
																			<m:cn>2</m:cn>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:divide/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:partialdiff/>
																		<m:ci>k</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:partialdiff/>
																		<m:apply>
																			<m:csymbol cd="ambiguous">superscript</m:csymbol>
																			<m:ci>λ</m:ci>
																			<m:ci>k</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:interval closure="open"/>
																		<m:ci>λ</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
																<m:ci/>
															</m:apply>
															<m:apply>
																<m:eq/>
																<m:ci>λ</m:ci>
																<m:cn>0</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">55</span>)</td>
								</tr>

							</table>
							<p class="p">The identities</p>
							<table class="equation" id="S6.E56">


								<tr valign="baseline" class="equation" id="S6.E56">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msub>
														<m:mfenced open="[" close="]">
															<m:mi>m</m:mi>
														</m:mfenced>
														<m:mrow>
															<m:mi>G</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:msup>
																		<m:mi>p</m:mi>
																		<m:mi>k</m:mi>
																	</m:msup>
																	<m:mo>,</m:mo>
																	<m:msup>
																		<m:mi>q</m:mi>
																		<m:mi>k</m:mi>
																	</m:msup>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:msub>
													<m:mo>=</m:mo>
													<m:mfrac>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mi>k</m:mi>
																	<m:mo>⁢</m:mo>
																	<m:mi>m</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mrow>
																<m:mi>G</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>p</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>q</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:msub>
														<m:msub>
															<m:mfenced open="[" close="]">
																<m:mi>k</m:mi>
															</m:mfenced>
															<m:mrow>
																<m:mi>G</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>p</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>q</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:msub>
													</m:mfrac>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:ci>m</m:ci>
															<m:apply>
																<m:times/>
																<m:ci>G</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>p</m:ci>
																		<m:ci>k</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:ci>q</m:ci>
																		<m:ci>k</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:divide/>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:apply>
																	<m:times/>
																	<m:ci>k</m:ci>
																	<m:ci>m</m:ci>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:ci>G</m:ci>
																	<m:apply>
																		<m:interval closure="open"/>
																		<m:ci>p</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>k</m:ci>
																<m:apply>
																	<m:times/>
																	<m:ci>G</m:ci>
																	<m:apply>
																		<m:interval closure="open"/>
																		<m:ci>p</m:ci>
																		<m:ci>q</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">56</span>)</td>
								</tr>

							</table>
							<p class="p">and</p>
							<table class="equation" id="S6.E57">


								<tr valign="baseline" class="equation" id="S6.E57">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:msub>
														<m:mfenced open="[" close="]">
															<m:mrow>
																<m:mi>k</m:mi>
																<m:mo>⁢</m:mo>
																<m:mi>m</m:mi>
															</m:mrow>
														</m:mfenced>
														<m:mrow>
															<m:mi>G</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>p</m:mi>
																	<m:mo>,</m:mo>
																	<m:mi>q</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:msub>
													<m:mo>=</m:mo>
													<m:mrow>
														<m:mover>
															<m:munder>
																<m:mo movablelimits="false">∑</m:mo>
																<m:mrow>
																	<m:mi>i</m:mi>
																	<m:mo movablelimits="false">=</m:mo>
																	<m:mn>1</m:mn>
																</m:mrow>
															</m:munder>
															<m:mi>k</m:mi>
														</m:mover>
														<m:mfenced open="(" close=")">
															<m:mtable rowspacing="0.2ex" columnspacing="0.4em">
																<m:mtr>
																	<m:mtd/>
																	<m:mtd columnalign="center">
																		<m:mi>k</m:mi>
																	</m:mtd>
																	<m:mtd/>
																</m:mtr>
																<m:mtr>
																	<m:mtd/>
																	<m:mtd columnalign="center">
																		<m:mi>i</m:mi>
																	</m:mtd>
																	<m:mtd/>
																</m:mtr>
															</m:mtable>
														</m:mfenced>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mfenced open="(" close=")">
																<m:mrow>
																	<m:mi>q</m:mi>
																	<m:mo>-</m:mo>
																	<m:mi>p</m:mi>
																</m:mrow>
															</m:mfenced>
															<m:mrow>
																<m:mi>i</m:mi>
																<m:mo>-</m:mo>
																<m:mn>1</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo>⁢</m:mo>
														<m:msubsup>
															<m:mfenced open="[" close="]">
																<m:mi>m</m:mi>
															</m:mfenced>
															<m:mrow>
																<m:mi>G</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>p</m:mi>
																		<m:mo>,</m:mo>
																		<m:mi>q</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mi>i</m:mi>
														</m:msubsup>
														<m:mo>⁢</m:mo>
														<m:msup>
															<m:mi>p</m:mi>
															<m:mrow>
																<m:mi>m</m:mi>
																<m:mo>⁢</m:mo>
																<m:mfenced open="(" close=")">
																	<m:mrow>
																		<m:mi>k</m:mi>
																		<m:mo>-</m:mo>
																		<m:mi>i</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:msup>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:apply>
															<m:csymbol cd="ambiguous">subscript</m:csymbol>
															<m:apply>
																<m:times/>
																<m:ci>k</m:ci>
																<m:ci>m</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:ci>G</m:ci>
																<m:apply>
																	<m:interval closure="open"/>
																	<m:ci>p</m:ci>
																	<m:ci>q</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:apply>
																	<m:csymbol cd="ambiguous">subscript</m:csymbol>
																	<m:sum/>
																	<m:apply>
																		<m:eq/>
																		<m:ci>i</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>k</m:ci>
															</m:apply>
															<m:apply>
																<m:times/>
																<m:mtext>ki</m:mtext>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:minus/>
																		<m:ci>q</m:ci>
																		<m:ci>p</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:minus/>
																		<m:ci>i</m:ci>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:apply>
																		<m:csymbol cd="ambiguous">subscript</m:csymbol>
																		<m:ci>m</m:ci>
																		<m:apply>
																			<m:times/>
																			<m:ci>G</m:ci>
																			<m:apply>
																				<m:interval closure="open"/>
																				<m:ci>p</m:ci>
																				<m:ci>q</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:ci>i</m:ci>
																</m:apply>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>p</m:ci>
																	<m:apply>
																		<m:times/>
																		<m:ci>m</m:ci>
																		<m:apply>
																			<m:minus/>
																			<m:ci>k</m:ci>
																			<m:ci>i</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">57</span>)</td>
								</tr>

							</table>
							<p class="p">are found to be useful when implementing Eq. (<a href="#S6.E55" title="Eq.55 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">55</a>).
(To avoid possible confusion we point out that the symbol
appearing in Eq. (<a href="#S6.E57" title="Eq.57 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">57</a>) is the conventional
binomial coefficient.)
Note that for the conventional bosons, for which <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:mi>q</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci/>
												<m:ci>p</m:ci>
												<m:eq/>
												<m:ci>q</m:ci>
												<m:eq/>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
Eq. (<a href="#S6.E55" title="Eq.55 in §6. A generating function for the deformed Stirling numbers&#10;of the first kind in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">55</a>) reduces to an expression <cite class="cite">[<a href="#bib.bib15" title="" class="ref">15</a>]</cite>
which can be related to the well-known generating
function for the conventional Stirling numbers of the first
kind <cite class="cite">[<a href="#bib.bib5" title="" class="ref">5</a>]</cite>.</p>
						</div>

					</div>

					<div class="section" id="S7">
						<h2 class="title section-title">
							<span class="refnum">
								<span class="reftype">§ </span>7. </span>Discussion</h2>
						<div class="para" id="S7.p1">
							<p class="p">In the present article we found that the normal ordering formulae
for powers of the boson number operator can be extended in a
simple and natural way to the M-type bosons, which satisfy
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mrow>
														<m:mi>a</m:mi>
														<m:mo>,</m:mo>
														<m:msup>
															<m:mi>a</m:mi>
															<m:mo>†</m:mo>
														</m:msup>
													</m:mrow>
												</m:mfenced>
												<m:mi>q</m:mi>
											</m:msub>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:interval closure="closed"/>
														<m:ci>a</m:ci>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
													</m:apply>
													<m:ci>q</m:ci>
												</m:apply>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. However, for the P-type bosons,
which satisfy
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:msub>
												<m:mfenced open="[" close="]">
													<m:mrow>
														<m:mi>a</m:mi>
														<m:mo>,</m:mo>
														<m:msup>
															<m:mi>a</m:mi>
															<m:mo>†</m:mo>
														</m:msup>
													</m:mrow>
												</m:mfenced>
												<m:mi>q</m:mi>
											</m:msub>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mrow>
													<m:mo>-</m:mo>
													<m:mover accent="true">
														<m:mi>n</m:mi>
														<m:mo>⌃</m:mo>
													</m:mover>
												</m:mrow>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:csymbol cd="ambiguous">subscript</m:csymbol>
													<m:apply>
														<m:interval closure="closed"/>
														<m:ci>a</m:ci>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
													</m:apply>
													<m:ci>q</m:ci>
												</m:apply>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:minus/>
														<m:apply>
															<m:ci>⌃</m:ci>
															<m:ci>n</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, as well as for the more general
G-type bosons, we found that the extension of the conventional
boson analysis results in “normal-ordering” expressions with
								<m:math display="inline">
									<m:semantics>
										<m:mover accent="true">
											<m:mi>n</m:mi>
											<m:mo>⌃</m:mo>
										</m:mover>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:ci>⌃</m:ci>
												<m:ci>n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-dependent coefficients.</p>
						</div>

						<div class="para" id="S7.p2">
							<p class="p">The marked difference between the M-type bosons and all the others
has already been noted before, in the context of the extension of the
Campbell-Baker-Hausdorff formula for products of exponential
operators <cite class="cite">[<a href="#bib.bib16" title="" class="ref">16</a>]</cite>. While the observations pointed out
above set apart the M-type bosons, the following may be taken to set
apart the P-type bosons disfavourably,
within the general set of G-type bosons:
Taking the Hamiltonian of the deformed harmonic oscillator to be
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi mathvariant="script">H</m:mi>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mfrac>
													<m:mrow>
														<m:mi mathvariant="normal">ℏ</m:mi>
														<m:mo>⁢</m:mo>
														<m:msub>
															<m:mi>ω</m:mi>
															<m:mn>0</m:mn>
														</m:msub>
													</m:mrow>
													<m:mn>2</m:mn>
												</m:mfrac>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mrow>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mo>†</m:mo>
															</m:msup>
															<m:mo>⁢</m:mo>
															<m:mi>a</m:mi>
														</m:mrow>
														<m:mo>+</m:mo>
														<m:mrow>
															<m:mi>a</m:mi>
															<m:mo>⁢</m:mo>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mo>†</m:mo>
															</m:msup>
														</m:mrow>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>H</m:ci>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:divide/>
														<m:apply>
															<m:times/>
															<m:ci>ℏ</m:ci>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>ω</m:ci>
																<m:cn>0</m:cn>
															</m:apply>
														</m:apply>
														<m:cn>2</m:cn>
													</m:apply>
													<m:apply>
														<m:plus/>
														<m:apply>
															<m:times/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>†</m:ci>
															</m:apply>
															<m:ci>a</m:ci>
														</m:apply>
														<m:apply>
															<m:times/>
															<m:ci>a</m:ci>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>†</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
and expanding in
powers of <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>s</m:mi>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mi>ln</m:mi>
												<m:mo>⁡</m:mo>
												<m:mi>q</m:mi>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>s</m:ci>
												<m:apply>
													<m:ln/>
													<m:ci>q</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>t</m:mi>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mi>ln</m:mi>
												<m:mo>⁡</m:mo>
												<m:mi>p</m:mi>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>t</m:ci>
												<m:apply>
													<m:ln/>
													<m:ci>p</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> (which we assume to be sufficiently
small), we find that</p>
							<table class="equation" id="S7.E58">


								<tr valign="baseline" class="equation" id="S7.E58">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mi mathvariant="script">H</m:mi>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mi mathvariant="normal">ℏ</m:mi>
															<m:mo>⁢</m:mo>
															<m:msub>
																<m:mi>ω</m:mi>
																<m:mn>0</m:mn>
															</m:msub>
															<m:mo>⁢</m:mo>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mfrac>
																		<m:mrow>
																			<m:mi>s</m:mi>
																			<m:mo>+</m:mo>
																			<m:mi>t</m:mi>
																		</m:mrow>
																		<m:mn>8</m:mn>
																	</m:mfrac>
																	<m:mo>+</m:mo>
																	<m:mrow>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mn>1</m:mn>
																				<m:mo>-</m:mo>
																				<m:mfrac>
																					<m:mrow>
																						<m:mi>s</m:mi>
																						<m:mo>+</m:mo>
																						<m:mi>t</m:mi>
																					</m:mrow>
																					<m:mn>2</m:mn>
																				</m:mfrac>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mover accent="true">
																					<m:mi>n</m:mi>
																					<m:mo>⌃</m:mo>
																				</m:mover>
																				<m:mo>+</m:mo>
																				<m:mfrac>
																					<m:mn>1</m:mn>
																					<m:mn>2</m:mn>
																				</m:mfrac>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>+</m:mo>
																	<m:mrow>
																		<m:mfrac>
																			<m:mrow>
																				<m:mi>s</m:mi>
																				<m:mo>+</m:mo>
																				<m:mi>t</m:mi>
																			</m:mrow>
																			<m:mn>2</m:mn>
																		</m:mfrac>
																		<m:mo>⁢</m:mo>
																		<m:msup>
																			<m:mfenced open="(" close=")">
																				<m:mrow>
																					<m:mover accent="true">
																						<m:mi>n</m:mi>
																						<m:mo>⌃</m:mo>
																					</m:mover>
																					<m:mo>+</m:mo>
																					<m:mfrac>
																						<m:mn>1</m:mn>
																						<m:mn>2</m:mn>
																					</m:mfrac>
																				</m:mrow>
																			</m:mfenced>
																			<m:mn>2</m:mn>
																		</m:msup>
																	</m:mrow>
																	<m:mo>+</m:mo>
																	<m:mi mathvariant="normal">⋯</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>.</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:eq/>
														<m:ci>H</m:ci>
														<m:apply>
															<m:times/>
															<m:ci>ℏ</m:ci>
															<m:apply>
																<m:csymbol cd="ambiguous">subscript</m:csymbol>
																<m:ci>ω</m:ci>
																<m:cn>0</m:cn>
															</m:apply>
															<m:apply>
																<m:plus/>
																<m:apply>
																	<m:divide/>
																	<m:apply>
																		<m:plus/>
																		<m:ci>s</m:ci>
																		<m:ci>t</m:ci>
																	</m:apply>
																	<m:cn>8</m:cn>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:minus/>
																		<m:cn>1</m:cn>
																		<m:apply>
																			<m:divide/>
																			<m:apply>
																				<m:plus/>
																				<m:ci>s</m:ci>
																				<m:ci>t</m:ci>
																			</m:apply>
																			<m:cn>2</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:plus/>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																		<m:apply>
																			<m:divide/>
																			<m:cn>1</m:cn>
																			<m:cn>2</m:cn>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:plus/>
																			<m:ci>s</m:ci>
																			<m:ci>t</m:ci>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																	<m:apply>
																		<m:csymbol cd="ambiguous">superscript</m:csymbol>
																		<m:apply>
																			<m:plus/>
																			<m:apply>
																				<m:ci>⌃</m:ci>
																				<m:ci>n</m:ci>
																			</m:apply>
																			<m:apply>
																				<m:divide/>
																				<m:cn>1</m:cn>
																				<m:cn>2</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>⋯</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">58</span>)</td>
								</tr>

							</table>
							<p class="p">Apart from an irrelevant shift of the energy zero and a renormalization
of the frequency into <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>ω</m:mi>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:msub>
													<m:mi>ω</m:mi>
													<m:mn>0</m:mn>
												</m:msub>
												<m:mo>⁢</m:mo>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:mn>1</m:mn>
														<m:mo>-</m:mo>
														<m:mfrac>
															<m:mrow>
																<m:mi>s</m:mi>
																<m:mo>+</m:mo>
																<m:mi>t</m:mi>
															</m:mrow>
															<m:mn>2</m:mn>
														</m:mfrac>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>ω</m:ci>
												<m:apply>
													<m:times/>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>ω</m:ci>
														<m:cn>0</m:cn>
													</m:apply>
													<m:apply>
														<m:minus/>
														<m:cn>1</m:cn>
														<m:apply>
															<m:divide/>
															<m:apply>
																<m:plus/>
																<m:ci>s</m:ci>
																<m:ci>t</m:ci>
															</m:apply>
															<m:cn>2</m:cn>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
this Hamiltonian contains a quadratic anharmonicity
unless <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>s</m:mi>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mo>-</m:mo>
												<m:mi>t</m:mi>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>s</m:ci>
												<m:apply>
													<m:minus/>
													<m:ci>t</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, i.e., unless <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mrow>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:minus/>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
It is true that a quadratic anharmonicity will emerge even for the
P-type oscillator
(<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>p</m:mi>
											<m:mo>=</m:mo>
											<m:msup>
												<m:mi>q</m:mi>
												<m:mrow>
													<m:mo>-</m:mo>
													<m:mn>1</m:mn>
												</m:mrow>
											</m:msup>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>p</m:ci>
												<m:apply>
													<m:csymbol cd="ambiguous">superscript</m:csymbol>
													<m:ci>q</m:ci>
													<m:apply>
														<m:minus/>
														<m:cn>1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>) as a residue of the fourth order term, but
it will be associated with a fourth order anharmonicity which may
well be inconsistent with the experimental spectrum of some
system of interest, such as a diatomic molecule.</p>
						</div>

						<div class="para" id="S7.p3">
							<p class="p">We finally point out that a coordinate and a conjugate
momentum can be defined for the deformed oscillator by means of
the relations
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mover accent="true">
												<m:mi>x</m:mi>
												<m:mo>⌃</m:mo>
											</m:mover>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mfenced open="(" close=")">
													<m:mrow>
														<m:msup>
															<m:mi>a</m:mi>
															<m:mo>†</m:mo>
														</m:msup>
														<m:mo>+</m:mo>
														<m:mi>a</m:mi>
													</m:mrow>
												</m:mfenced>
												<m:mo>/</m:mo>
												<m:msqrt>
													<m:mn>2</m:mn>
												</m:msqrt>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:ci>⌃</m:ci>
													<m:ci>x</m:ci>
												</m:apply>
												<m:apply>
													<m:divide/>
													<m:apply>
														<m:plus/>
														<m:apply>
															<m:csymbol cd="ambiguous">superscript</m:csymbol>
															<m:ci>a</m:ci>
															<m:ci>†</m:ci>
														</m:apply>
														<m:ci>a</m:ci>
													</m:apply>
													<m:apply>
														<m:ci/>
														<m:cn>2</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and
								<m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mover accent="true">
												<m:mi>p</m:mi>
												<m:mo>⌃</m:mo>
											</m:mover>
											<m:mo>=</m:mo>
											<m:mrow>
												<m:mrow>
													<m:mi>i</m:mi>
													<m:mo>⁢</m:mo>
													<m:mfenced open="(" close=")">
														<m:mrow>
															<m:msup>
																<m:mi>a</m:mi>
																<m:mo>†</m:mo>
															</m:msup>
															<m:mo>-</m:mo>
															<m:mi>a</m:mi>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
												<m:mo>/</m:mo>
												<m:msqrt>
													<m:mn>2</m:mn>
												</m:msqrt>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:ci>⌃</m:ci>
													<m:ci>p</m:ci>
												</m:apply>
												<m:apply>
													<m:divide/>
													<m:apply>
														<m:times/>
														<m:ci>i</m:ci>
														<m:apply>
															<m:minus/>
															<m:apply>
																<m:csymbol cd="ambiguous">superscript</m:csymbol>
																<m:ci>a</m:ci>
																<m:ci>†</m:ci>
															</m:apply>
															<m:ci>a</m:ci>
														</m:apply>
													</m:apply>
													<m:apply>
														<m:ci/>
														<m:cn>2</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. Application of
Eq. (<a href="#S3.E20" title="Eq.20 in §3. Some algebraic properties of deformed boson operators in Normal ordering for deformed boson operators and&#10;operator-valued deformed Stirling numbers" class="ref">20</a>) with the choice <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mi>Q</m:mi>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:ci>Q</m:ci>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> results in
(for <m:math display="inline">
									<m:semantics>
										<m:mrow>
											<m:mrow>
												<m:mi mathvariant="normal">ℏ</m:mi>
												<m:mo>⁢</m:mo>
												<m:msub>
													<m:mi>ω</m:mi>
													<m:mn>0</m:mn>
												</m:msub>
											</m:mrow>
											<m:mo>=</m:mo>
											<m:mn>1</m:mn>
										</m:mrow>
										<m:annotation-xml encoding="MathML-Content">
											<m:apply>
												<m:eq/>
												<m:apply>
													<m:times/>
													<m:ci>ℏ</m:ci>
													<m:apply>
														<m:csymbol cd="ambiguous">subscript</m:csymbol>
														<m:ci>ω</m:ci>
														<m:cn>0</m:cn>
													</m:apply>
												</m:apply>
												<m:cn>1</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>)</p>
							<table class="equation" id="S7.E59">


								<tr valign="baseline" class="equation" id="S7.E59">
									<td class="eqpad"/>
									<td nowrap="yes" align="center" colspan="1">
										<m:math display="block">
											<m:semantics>
												<m:mrow>
													<m:mrow>
														<m:mfenced open="[" close="]">
															<m:mrow>
																<m:mover accent="true">
																	<m:mi>x</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
																<m:mo>,</m:mo>
																<m:mover accent="true">
																	<m:mi>p</m:mi>
																	<m:mo>⌃</m:mo>
																</m:mover>
															</m:mrow>
														</m:mfenced>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mi>i</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mi>a</m:mi>
																	<m:mo>,</m:mo>
																	<m:msup>
																		<m:mi>a</m:mi>
																		<m:mo>†</m:mo>
																	</m:msup>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:mo>=</m:mo>
														<m:mrow>
															<m:mi>i</m:mi>
															<m:mo>⁢</m:mo>
															<m:mfenced open="[" close="]">
																<m:mrow>
																	<m:mn>1</m:mn>
																	<m:mo>+</m:mo>
																	<m:mrow>
																		<m:none/>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mi>s</m:mi>
																				<m:mo>+</m:mo>
																				<m:mi>t</m:mi>
																				<m:mo>-</m:mo>
																				<m:mrow>
																					<m:mfrac>
																						<m:msup>
																							<m:mfenced open="(" close=")">
																								<m:mrow>
																									<m:mi>s</m:mi>
																									<m:mo>+</m:mo>
																									<m:mi>t</m:mi>
																								</m:mrow>
																							</m:mfenced>
																							<m:mn>2</m:mn>
																						</m:msup>
																						<m:mn>2</m:mn>
																					</m:mfrac>
																					<m:mo>⁢</m:mo>
																					<m:none/>
																				</m:mrow>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo>⁢</m:mo>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																	</m:mrow>
																	<m:mo>+</m:mo>
																	<m:mrow>
																		<m:mfrac>
																			<m:mrow>
																				<m:msup>
																					<m:mi>s</m:mi>
																					<m:mn>2</m:mn>
																				</m:msup>
																				<m:mo>+</m:mo>
																				<m:mrow>
																					<m:mi>s</m:mi>
																					<m:mo>⁢</m:mo>
																					<m:mi>t</m:mi>
																				</m:mrow>
																				<m:mo>+</m:mo>
																				<m:msup>
																					<m:mi>t</m:mi>
																					<m:mn>2</m:mn>
																				</m:msup>
																			</m:mrow>
																			<m:mn>2</m:mn>
																		</m:mfrac>
																		<m:mo>⁢</m:mo>
																		<m:mover accent="true">
																			<m:mi>n</m:mi>
																			<m:mo>⌃</m:mo>
																		</m:mover>
																		<m:mo>⁢</m:mo>
																		<m:mfenced open="(" close=")">
																			<m:mrow>
																				<m:mover accent="true">
																					<m:mi>n</m:mi>
																					<m:mo>⌃</m:mo>
																				</m:mover>
																				<m:mo>+</m:mo>
																				<m:mn>1</m:mn>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo>+</m:mo>
																	<m:mi mathvariant="normal">⋯</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:mo>,</m:mo>
												</m:mrow>
												<m:annotation-xml encoding="MathML-Content">
													<m:apply>
														<m:ci/>
														<m:apply>
															<m:interval closure="closed"/>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>x</m:ci>
															</m:apply>
															<m:apply>
																<m:ci>⌃</m:ci>
																<m:ci>p</m:ci>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>i</m:ci>
															<m:apply>
																<m:interval closure="closed"/>
																<m:ci>a</m:ci>
																<m:apply>
																	<m:csymbol cd="ambiguous">superscript</m:csymbol>
																	<m:ci>a</m:ci>
																	<m:ci>†</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq/>
														<m:apply>
															<m:times/>
															<m:ci>i</m:ci>
															<m:apply>
																<m:plus/>
																<m:cn>1</m:cn>
																<m:apply>
																	<m:times/>
																	<m:ci/>
																	<m:apply>
																		<m:minus/>
																		<m:apply>
																			<m:plus/>
																			<m:ci>s</m:ci>
																			<m:ci>t</m:ci>
																		</m:apply>
																		<m:apply>
																			<m:times/>
																			<m:apply>
																				<m:divide/>
																				<m:apply>
																					<m:csymbol cd="ambiguous">superscript</m:csymbol>
																					<m:apply>
																						<m:plus/>
																						<m:ci>s</m:ci>
																						<m:ci>t</m:ci>
																					</m:apply>
																					<m:cn>2</m:cn>
																				</m:apply>
																				<m:cn>2</m:cn>
																			</m:apply>
																			<m:ci/>
																		</m:apply>
																	</m:apply>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply>
																	<m:times/>
																	<m:apply>
																		<m:divide/>
																		<m:apply>
																			<m:plus/>
																			<m:apply>
																				<m:csymbol cd="ambiguous">superscript</m:csymbol>
																				<m:ci>s</m:ci>
																				<m:cn>2</m:cn>
																			</m:apply>
																			<m:apply>
																				<m:times/>
																				<m:ci>s</m:ci>
																				<m:ci>t</m:ci>
																			</m:apply>
																			<m:apply>
																				<m:csymbol cd="ambiguous">superscript</m:csymbol>
																				<m:ci>t</m:ci>
																				<m:cn>2</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn>2</m:cn>
																	</m:apply>
																	<m:apply>
																		<m:ci>⌃</m:ci>
																		<m:ci>n</m:ci>
																	</m:apply>
																	<m:apply>
																		<m:plus/>
																		<m:apply>
																			<m:ci>⌃</m:ci>
																			<m:ci>n</m:ci>
																		</m:apply>
																		<m:cn>1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci>⋯</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td class="eqpad"/>
									<td rowspan="1" nowrap="yes" valign="middle" align="right">(<span class="refnum">59</span>)</td>
								</tr>

							</table>
							<p class="p">from which follows the deformed uncertainty relation.</p>
						</div>

						<div class="para" id="S7.p4">
							<p class="p">
								<span style="" class="text bold">Acknowledgements</span>
							</p>
						</div>

						<div class="para" id="S7.p5">
							<p class="p">One of the authors (JK) would like to thank the
Région Rhône-Alpes for a visiting fellowship
and the Institut
de Physique Nucléaire de Lyon for its kind hospitality.</p>
						</div>

					</div>

					<div class="bibliography" id="bib">
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					</div>

				</div>

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